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Metamath Proof Explorer

Theorem brgic

Description: The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015)

		Ref	Expression
	Assertion	brgic	$⊢ R ≃_{𝑔} S \leftrightarrow R GrpIso S \neq \emptyset$

Step	Hyp	Ref	Expression
1		df-gic	$⊢ ≃_{𝑔} = {GrpIso}^{-1} [V ∖ 1_{𝑜}]$
2		gimfn	$⊢ GrpIso Fn (Grp \times Grp)$
3	1 2	brwitnlem	$⊢ R ≃_{𝑔} S \leftrightarrow R GrpIso S \neq \emptyset$