Theorem invinv

Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of Adamek p. 29. (Contributed by Mario Carneiro, 2-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		invfval.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		invfval.n	⊢ 𝑁 = ( Inv ‘ 𝐶 )
3		invfval.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
4		invss.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5		invss.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6		isoval.n	⊢ 𝐼 = ( Iso ‘ 𝐶 )
7		invinv.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
8	1 2 3 4 5	invsym2	⊢ ( 𝜑 → ◡ ( 𝑋 𝑁 𝑌 ) = ( 𝑌 𝑁 𝑋 ) )
9	8	fveq1d	⊢ ( 𝜑 → ( ◡ ( 𝑋 𝑁 𝑌 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = ( ( 𝑌 𝑁 𝑋 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) )
10	1 2 3 4 5 6	invf1o	⊢ ( 𝜑 → ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) )
11		f1ocnvfv1	⊢ ( ( ( 𝑋 𝑁 𝑌 ) : ( 𝑋 𝐼 𝑌 ) –1-1-onto→ ( 𝑌 𝐼 𝑋 ) ∧ 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) ) → ( ◡ ( 𝑋 𝑁 𝑌 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )
12	10 7 11	syl2anc	⊢ ( 𝜑 → ( ◡ ( 𝑋 𝑁 𝑌 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )
13	9 12	eqtr3d	⊢ ( 𝜑 → ( ( 𝑌 𝑁 𝑋 ) ‘ ( ( 𝑋 𝑁 𝑌 ) ‘ 𝐹 ) ) = 𝐹 )