Theorem gimf1o

Description: An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

	Ref	Expression
Hypotheses	isgim.b	$$⊢B={\mathrm{Base}}_R$$
	isgim.c	$$⊢C={\mathrm{Base}}_S$$
Assertion	gimf1o	$$⊢F\in \left(R\mathrm{GrpIso}S\right)\to F:B\underset{1-1\; onto}{⟶}C$$

Proof

Step	Hyp	Ref	Expression
1		isgim.b	$$⊢B={\mathrm{Base}}_R$$
2		isgim.c	$$⊢C={\mathrm{Base}}_S$$
3	1 2	isgim	$$⊢F\in \left(R\mathrm{GrpIso}S\right)\leftrightarrow \left(F\in \left(R\mathrm{GrpHom}S\right)\wedge F:B\underset{1-1\; onto}{⟶}C\right)$$
4	3	simprbi	$$⊢F\in \left(R\mathrm{GrpIso}S\right)\to F:B\underset{1-1\; onto}{⟶}C$$