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Metamath Proof Explorer

Theorem homfeqd

Description: If two structures have the same Hom slot, they have the same Hom-sets. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses homfeqd.1 ⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
homfeqd.2 ⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐷 ) )
Assertion homfeqd ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 homfeqd.1 ⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐷 ) )
2 homfeqd.2 ⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐷 ) )
3 2 oveqd ⊢ ( 𝜑 → ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
4 3 ralrimivw ⊢ ( 𝜑 → ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
5 4 ralrimivw ⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) )
6 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
7 eqid ⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
8 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) )
9 6 7 8 1 homfeq ⊢ ( 𝜑 → ( ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) = ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) ) )
10 5 9 mpbird ⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐷 ) )