Theorem gimghm

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

	Ref	Expression
Assertion	gimghm	$$⊢F\in \left(R\mathrm{GrpIso}S\right)\to F\in \left(R\mathrm{GrpHom}S\right)$$

Proof

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