Theorem rimf1o

Description: An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019)

	Ref	Expression
Hypotheses	rhmf1o.b	|- B = ( Base ` R )
	rhmf1o.c	|- C = ( Base ` S )
Assertion	rimf1o	|- ( F e. ( R RingIso S ) -> F : B -1-1-onto-> C )

Proof

Step	Hyp	Ref	Expression
1		rhmf1o.b	|- B = ( Base ` R )
2		rhmf1o.c	|- C = ( Base ` S )
3		rimrcl	|- ( F e. ( R RingIso S ) -> ( R e. _V /\ S e. _V ) )
4	1 2	isrim	|- ( ( R e. _V /\ S e. _V ) -> ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) ) )
5		simpr	|- ( ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) -> F : B -1-1-onto-> C )
6	4 5	syl6bi	|- ( ( R e. _V /\ S e. _V ) -> ( F e. ( R RingIso S ) -> F : B -1-1-onto-> C ) )
7	3 6	mpcom	|- ( F e. ( R RingIso S ) -> F : B -1-1-onto-> C )