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Metamath Proof Explorer

Theorem catpropd

Description: Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017)

Ref Expression
Hypotheses catpropd.1 ⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
catpropd.2 ⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
catpropd.3 ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
catpropd.4 ⊢ ( 𝜑 → 𝐷 ∈ 𝑊 )
Assertion catpropd ( 𝜑 → ( 𝐶 ∈ Cat ↔ 𝐷 ∈ Cat ) )

Proof

Step Hyp Ref Expression
1 catpropd.1 ⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
2 catpropd.2 ⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
3 catpropd.3 ⊢ ( 𝜑 → 𝐶 ∈ 𝑉 )
4 catpropd.4 ⊢ ( 𝜑 → 𝐷 ∈ 𝑊 )
5 simpl ⊢ ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
6 5 2ralimi ⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
7 6 2ralimi ⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
8 7 adantl ⊢ ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
9 8 ralimi ⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
10 9 a1i ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
11 simpl ⊢ ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
12 11 2ralimi ⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
13 12 2ralimi ⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
14 13 adantl ⊢ ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
15 14 ralimi ⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
16 15 a1i ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
17 nfra1 ⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
18 nfv ⊢ Ⅎ 𝑥 ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 )
19 nfra1 ⊢ Ⅎ 𝑧 ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
20 nfv ⊢ Ⅎ 𝑦 ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 )
21 nfra1 ⊢ Ⅎ 𝑔 ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
22 nfv ⊢ Ⅎ 𝑓 ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 )
23 oveq1 ⊢ ( 𝑔 = ℎ → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) )
24 23 eleq1d ⊢ ( 𝑔 = ℎ → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
25 24 cbvralvw ⊢ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
26 oveq2 ⊢ ( 𝑓 = 𝑔 → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) )
27 26 eleq1d ⊢ ( 𝑓 = 𝑔 → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
28 27 ralbidv ⊢ ( 𝑓 = 𝑔 → ( ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
29 25 28 bitrid ⊢ ( 𝑓 = 𝑔 → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
30 21 22 29 cbvralw ⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
31 oveq2 ⊢ ( 𝑧 = 𝑤 → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
32 oveq2 ⊢ ( 𝑧 = 𝑤 → ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) = ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) )
33 32 oveqd ⊢ ( 𝑧 = 𝑤 → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) = ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) )
34 oveq2 ⊢ ( 𝑧 = 𝑤 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) )
35 33 34 eleq12d ⊢ ( 𝑧 = 𝑤 → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
36 31 35 raleqbidv ⊢ ( 𝑧 = 𝑤 → ( ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
37 36 ralbidv ⊢ ( 𝑧 = 𝑤 → ( ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
38 30 37 bitrid ⊢ ( 𝑧 = 𝑤 → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
39 38 cbvralvw ⊢ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) )
40 oveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
41 oveq1 ⊢ ( 𝑦 = 𝑧 → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) = ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) )
42 opeq2 ⊢ ( 𝑦 = 𝑧 → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝑥 , 𝑧 ⟩ )
43 42 oveq1d ⊢ ( 𝑦 = 𝑧 → ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) = ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) )
44 43 oveqd ⊢ ( 𝑦 = 𝑧 → ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) )
45 44 eleq1d ⊢ ( 𝑦 = 𝑧 → ( ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
46 41 45 raleqbidv ⊢ ( 𝑦 = 𝑧 → ( ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
47 40 46 raleqbidv ⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
48 47 ralbidv ⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ ℎ ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
49 39 48 bitrid ⊢ ( 𝑦 = 𝑧 → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
50 19 20 49 cbvralw ⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) )
51 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) )
52 opeq1 ⊢ ( 𝑥 = 𝑦 → ⟨ 𝑥 , 𝑧 ⟩ = ⟨ 𝑦 , 𝑧 ⟩ )
53 52 oveq1d ⊢ ( 𝑥 = 𝑦 → ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) = ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) )
54 53 oveqd ⊢ ( 𝑥 = 𝑦 → ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) = ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) )
55 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) = ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
56 54 55 eleq12d ⊢ ( 𝑥 = 𝑦 → ( ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
57 56 ralbidv ⊢ ( 𝑥 = 𝑦 → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
58 51 57 raleqbidv ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
59 58 ralbidv ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
60 ralcom ⊢ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
61 59 60 bitrdi ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
62 61 ralbidv ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑤 ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
63 50 62 bitrid ⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) )
64 17 18 63 cbvralw ⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
65 64 biimpi ⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
66 65 ancri ⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
67 r19.26 ⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
68 r19.26 ⊢ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ↔ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
69 r19.26 ⊢ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ↔ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) )
70 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
71 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
72 eqid ⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
73 eqid ⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 )
74 1 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
75 74 ad4antr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
76 75 ad4antr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
77 2 ad5antr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
78 77 ad4antr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
79 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
80 79 ad2antrr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
81 80 ad4antr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
82 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
83 82 ad4antr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
84 simpllr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑤 ∈ ( Base ‘ 𝐶 ) )
85 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
86 85 ad4antr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
87 simpr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) )
88 70 71 72 73 76 78 81 83 84 86 87 comfeqval ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) )
89 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
90 89 ad4antr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
91 simp-4r ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) )
92 simplr ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) )
93 70 71 72 73 76 78 81 90 84 91 92 comfeqval ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) )
94 88 93 eqeq12d ⊢ ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
95 94 ex ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
96 95 ralimdva ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
97 ralbi ⊢ ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
98 96 97 syl6 ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
99 98 ralimdva ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) → ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
100 99 impancom ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
101 100 impr ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
102 ralbi ⊢ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
103 101 102 syl ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) )
104 103 anbi2d ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
105 104 ex ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
106 105 ralimdva ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
107 69 106 biimtrrid ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
108 107 expdimp ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
109 ralbi ⊢ ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
110 108 109 syl6 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
111 110 an32s ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
112 111 ralimdva ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
113 ralbi ⊢ ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
114 112 113 syl6 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
115 114 expimpd ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
116 115 ralimdva ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
117 ralbi ⊢ ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
118 116 117 syl6 ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
119 68 118 biimtrrid ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
120 119 ralimdva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
121 ralbi ⊢ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
122 120 121 syl6 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
123 67 122 biimtrrid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
124 123 imp ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
125 124 an4s ⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) )
126 125 anbi2d ⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) ) → ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
127 126 expr ⊢ ( ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
128 127 ralimdva ⊢ ( ( 𝜑 ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
129 128 expimpd ⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑤 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
130 ralbi ⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
131 66 129 130 syl56 ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) ) )
132 10 16 131 pm5.21ndd ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
133 1 homfeqbas ⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
134 eqid ⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
135 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
136 70 71 134 74 135 135 homfeqval ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) )
137 133 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
138 74 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
139 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
140 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
141 70 71 134 138 139 140 homfeqval ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) )
142 1 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
143 2 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
144 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
145 simp-4r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
146 simpr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) )
147 simpllr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) )
148 70 71 72 73 142 143 144 145 145 146 147 comfeqval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) )
149 148 eqeq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ↔ ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ) )
150 141 149 raleqbidva ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ) )
151 70 71 134 138 140 139 homfeqval ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
152 1 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
153 2 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
154 simp-4r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
155 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
156 simpllr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) )
157 simpr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
158 70 71 72 73 152 153 154 154 155 156 157 comfeqval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) )
159 158 eqeq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ↔ ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) )
160 151 159 raleqbidva ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) )
161 150 160 anbi12d ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
162 137 161 raleqbidva ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
163 136 162 rexeqbidva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ↔ ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ) )
164 133 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
165 164 adantr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
166 74 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
167 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
168 70 71 134 166 79 167 homfeqval ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
169 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
170 70 71 134 166 167 169 homfeqval ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
171 170 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) )
172 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) )
173 70 71 72 73 75 77 80 82 89 85 172 comfeqval ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
174 70 71 134 166 79 169 homfeqval ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) )
175 174 ad2antrr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) )
176 173 175 eleq12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ↔ ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ) )
177 164 ad4antr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
178 75 adantr ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
179 simp-4r ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
180 simpr ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → 𝑤 ∈ ( Base ‘ 𝐶 ) )
181 70 71 134 178 179 180 homfeqval ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) = ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) )
182 166 ad4antr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )
183 2 ad7antr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( compf ‘ 𝐶 ) = ( compf ‘ 𝐷 ) )
184 167 ad4antr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑦 ∈ ( Base ‘ 𝐶 ) )
185 169 ad4antr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑧 ∈ ( Base ‘ 𝐶 ) )
186 simplr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑤 ∈ ( Base ‘ 𝐶 ) )
187 simpllr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) )
188 simpr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) )
189 70 71 72 73 182 183 184 185 186 187 188 comfeqval ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) = ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) )
190 189 oveq1d ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) )
191 79 ad4antr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) )
192 simp-4r ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) )
193 70 71 72 73 182 183 191 184 185 192 187 comfeqval ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) )
194 193 oveq2d ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) )
195 190 194 eqeq12d ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) ∧ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ) → ( ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) )
196 181 195 raleqbidva ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) )
197 177 196 raleqbidva ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ↔ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) )
198 176 197 anbi12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) ∧ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ) → ( ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
199 171 198 raleqbidva ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ) → ( ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
200 168 199 raleqbidva ⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑧 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
201 165 200 raleqbidva ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
202 164 201 raleqbidva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) )
203 163 202 anbi12d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
204 133 203 raleqbidva ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
205 132 204 bitrd ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
206 70 71 72 iscat ⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
207 3 206 syl ⊢ ( 𝜑 → ( 𝐶 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐶 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ 𝑧 ∈ ( Base ‘ 𝐶 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐶 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐶 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐶 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐶 ) 𝑧 ) 𝑓 ) ) ) ) ) )
208 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
209 208 134 73 iscat ⊢ ( 𝐷 ∈ 𝑊 → ( 𝐷 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
210 4 209 syl ⊢ ( 𝜑 → ( 𝐷 ∈ Cat ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐷 ) ( ∃ 𝑔 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑥 ) ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ( ∀ 𝑓 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑥 ) ( 𝑔 ( ⟨ 𝑦 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑥 ) 𝑓 ) = 𝑓 ∧ ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ( 𝑓 ( ⟨ 𝑥 , 𝑥 ⟩ ( comp ‘ 𝐷 ) 𝑦 ) 𝑔 ) = 𝑓 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑧 ∈ ( Base ‘ 𝐷 ) ∀ 𝑓 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 ( Hom ‘ 𝐷 ) 𝑧 ) ( ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑧 ) ∧ ∀ 𝑤 ∈ ( Base ‘ 𝐷 ) ∀ ℎ ∈ ( 𝑧 ( Hom ‘ 𝐷 ) 𝑤 ) ( ( ℎ ( ⟨ 𝑦 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑔 ) ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) 𝑓 ) = ( ℎ ( ⟨ 𝑥 , 𝑧 ⟩ ( comp ‘ 𝐷 ) 𝑤 ) ( 𝑔 ( ⟨ 𝑥 , 𝑦 ⟩ ( comp ‘ 𝐷 ) 𝑧 ) 𝑓 ) ) ) ) ) )
211 205 207 210 3bitr4d ⊢ ( 𝜑 → ( 𝐶 ∈ Cat ↔ 𝐷 ∈ Cat ) )