Theorem gicsym

Description: Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Assertion	gicsym	$$⊢R{\simeq }_{𝑔}S\to S{\simeq }_{𝑔}R$$

Proof

Step	Hyp	Ref	Expression
1		brgic	$$⊢R{\simeq }_{𝑔}S\leftrightarrow R\mathrm{GrpIso}S\ne \varnothing$$
2		n0	$$⊢R\mathrm{GrpIso}S\ne \varnothing \leftrightarrow \exists ff\in \left(R\mathrm{GrpIso}S\right)$$
3		gimcnv	$$⊢f\in \left(R\mathrm{GrpIso}S\right)\to f^{-1}\in \left(S\mathrm{GrpIso}R\right)$$
4		brgici	$$⊢f^{-1}\in \left(S\mathrm{GrpIso}R\right)\to S{\simeq }_{𝑔}R$$
5	3 4	syl	$$⊢f\in \left(R\mathrm{GrpIso}S\right)\to S{\simeq }_{𝑔}R$$
6	5	exlimiv	$$⊢\exists ff\in \left(R\mathrm{GrpIso}S\right)\to S{\simeq }_{𝑔}R$$
7	2 6	sylbi	$$⊢R\mathrm{GrpIso}S\ne \varnothing \to S{\simeq }_{𝑔}R$$
8	1 7	sylbi	$$⊢R{\simeq }_{𝑔}S\to S{\simeq }_{𝑔}R$$