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Metamath Proof Explorer

Theorem catcrcl2

Description: Reverse closure for the category of categories (in a universe) (Contributed by Zhi Wang, 14-Nov-2025)

Ref Expression
Hypotheses catcrcl.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
catcrcl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
catcrcl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
catcrcl2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
Assertion catcrcl2 ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 catcrcl.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2 catcrcl.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
3 catcrcl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
4 catcrcl2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
5 1 2 3 catcrcl ⊢ ( 𝜑 → 𝑈 ∈ V )
6 1 4 5 2 catchomfval ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) )
7 6 oveqd ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) 𝑌 ) )
8 3 7 eleqtrd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) 𝑌 ) )
9 eqid ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) )
10 9 elmpocl ⊢ ( 𝐹 ∈ ( 𝑋 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) 𝑌 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )
11 8 10 syl ⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )