Theorem isrim

Description: An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019) Remove sethood antecedent. (Revised by SN, 12-Jan-2025)

	Ref	Expression
Hypotheses	rhmf1o.b	|- B = ( Base ` R )
	rhmf1o.c	|- C = ( Base ` S )
Assertion	isrim	|- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom 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		isrim0	|- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) )
4	1 2	rhmf1o	|- ( F e. ( R RingHom S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S RingHom R ) ) )
5	4	bicomd	|- ( F e. ( R RingHom S ) -> ( `' F e. ( S RingHom R ) <-> F : B -1-1-onto-> C ) )
6	5	pm5.32i	|- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) <-> ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) )
7	3 6	bitri	|- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ F : B -1-1-onto-> C ) )