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	$$⊢B={\mathrm{Base}}_C$$
	invfval.n	$$⊢N=\mathrm{Inv}\left(C\right)$$
	invfval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	invfval.x	$$⊢\phi \to X\in B$$
	invfval.y	$$⊢\phi \to Y\in B$$
	isoval.n	$$⊢I=\mathrm{Iso}\left(C\right)$$
	inviso1.1	$$⊢\phi \to F\left(XNY\right)G$$
Assertion	inviso1	$$⊢\phi \to F\in \left(XIY\right)$$

Proof

Step	Hyp	Ref	Expression
1		invfval.b	$$⊢B={\mathrm{Base}}_C$$
2		invfval.n	$$⊢N=\mathrm{Inv}\left(C\right)$$
3		invfval.c	$$⊢\phi \to C\in \mathrm{Cat}$$
4		invfval.x	$$⊢\phi \to X\in B$$
5		invfval.y	$$⊢\phi \to Y\in B$$
6		isoval.n	$$⊢I=\mathrm{Iso}\left(C\right)$$
7		inviso1.1	$$⊢\phi \to F\left(XNY\right)G$$
8	1 2 3 4 5	invfun	$$⊢\phi \to \mathrm{Fun}\left(XNY\right)$$
9		funrel	$$⊢\mathrm{Fun}\left(XNY\right)\to \mathrm{Rel}\left(XNY\right)$$
10	8 9	syl	$$⊢\phi \to \mathrm{Rel}\left(XNY\right)$$
11		releldm	$$⊢\left(\mathrm{Rel}\left(XNY\right)\wedge F\left(XNY\right)G\right)\to F\in \mathrm{dom}\left(XNY\right)$$
12	10 7 11	syl2anc	$$⊢\phi \to F\in \mathrm{dom}\left(XNY\right)$$
13	1 2 3 4 5 6	isoval	$$⊢\phi \to XIY=\mathrm{dom}\left(XNY\right)$$
14	12 13	eleqtrrd	$$⊢\phi \to F\in \left(XIY\right)$$