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Metamath Proof Explorer

Theorem isorcl2

Description: Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025)

Ref Expression
Hypotheses isorcl.i ⊢ 𝐼 = ( Iso ‘ 𝐶 )
isorcl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
isorcl2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
Assertion isorcl2 ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 isorcl.i ⊢ 𝐼 = ( Iso ‘ 𝐶 )
2 isorcl.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
3 isorcl2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
4 eqid ⊢ ( Inv ‘ 𝐶 ) = ( Inv ‘ 𝐶 )
5 1 2 isorcl ⊢ ( 𝜑 → 𝐶 ∈ Cat )
6 3 4 5 1 isofval2 ⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 ( Inv ‘ 𝐶 ) 𝑦 ) ) )
7 6 oveqd ⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) = ( 𝑋 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 ( Inv ‘ 𝐶 ) 𝑦 ) ) 𝑌 ) )
8 2 7 eleqtrd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 ( Inv ‘ 𝐶 ) 𝑦 ) ) 𝑌 ) )
9 eqid ⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 ( Inv ‘ 𝐶 ) 𝑦 ) ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 ( Inv ‘ 𝐶 ) 𝑦 ) )
10 9 elmpocl ⊢ ( 𝐹 ∈ ( 𝑋 ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ dom ( 𝑥 ( Inv ‘ 𝐶 ) 𝑦 ) ) 𝑌 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )
11 8 10 syl ⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )