Theorem lmimf1o

Description: An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015)

	Ref	Expression
Hypotheses	islmim.b	$$⊢B={\mathrm{Base}}_R$$
	islmim.c	$$⊢C={\mathrm{Base}}_S$$
Assertion	lmimf1o	$$⊢F\in \left(R\mathrm{LMIso}S\right)\to F:B\underset{1-1\; onto}{⟶}C$$

Proof

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