Theorem inviso1

Description: If G is an inverse to F , then F 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	inviso1	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )

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	invfun	⊢ ( 𝜑 → Fun ( 𝑋 𝑁 𝑌 ) )
9		funrel	⊢ ( Fun ( 𝑋 𝑁 𝑌 ) → Rel ( 𝑋 𝑁 𝑌 ) )
10	8 9	syl	⊢ ( 𝜑 → Rel ( 𝑋 𝑁 𝑌 ) )
11		releldm	⊢ ( ( Rel ( 𝑋 𝑁 𝑌 ) ∧ 𝐹 ( 𝑋 𝑁 𝑌 ) 𝐺 ) → 𝐹 ∈ dom ( 𝑋 𝑁 𝑌 ) )
12	10 7 11	syl2anc	⊢ ( 𝜑 → 𝐹 ∈ dom ( 𝑋 𝑁 𝑌 ) )
13	1 2 3 4 5 6	isoval	⊢ ( 𝜑 → ( 𝑋 𝐼 𝑌 ) = dom ( 𝑋 𝑁 𝑌 ) )
14	12 13	eleqtrrd	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )