Theorem inviso2

Description: If G is an inverse to F , then G is an isomorphism. (Contributed by Mario Carneiro, 3-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 )
	inviso1.1	⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 )
Assertion	inviso2	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐼 𝑋 ) )

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		inviso1.1	⊢ ( 𝜑 → 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 )
8	1 2 3 4 5	invsym	⊢ ( 𝜑 → ( 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ↔ 𝐺 ( 𝑌 𝑁 𝑋 ) 𝐹 ) )
9	7 8	mpbid	⊢ ( 𝜑 → 𝐺 ( 𝑌 𝑁 𝑋 ) 𝐹 )
10	1 2 3 5 4 6 9	inviso1	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐼 𝑋 ) )