Theorem invisoinvr

Description: The inverse of an isomorphism is invers to the isomorphism. (Contributed by AV, 9-Apr-2020)

	Ref	Expression
Hypotheses	invisoinv.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	invisoinv.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
	invisoinv.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
	invisoinv.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	invisoinv.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	invisoinv.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	invisoinv.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
Assertion	invisoinvr	⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		invisoinv.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		invisoinv.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
3		invisoinv.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
4		invisoinv.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
5		invisoinv.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		invisoinv.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		invisoinv.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
8	1 2 3 4 5 6 7	invisoinvl	⊢ ( 𝜑 → ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 𝑌 𝑁 𝑋 ) 𝐹 )
9	1 3 4 5 6	invsym	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ↔ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ( 𝑌 𝑁 𝑋 ) 𝐹 ) )
10	8 9	mpbird	⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) )