Theorem rngoiso1o

Description: A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Hypotheses	rngisoval.1	|- G = ( 1st ` R )
	rngisoval.2	|- X = ran G
	rngisoval.3	|- J = ( 1st ` S )
	rngisoval.4	|- Y = ran J
Assertion	rngoiso1o	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngIso S ) ) -> F : X -1-1-onto-> Y )

Proof

Step	Hyp	Ref	Expression
1		rngisoval.1	|- G = ( 1st ` R )
2		rngisoval.2	|- X = ran G
3		rngisoval.3	|- J = ( 1st ` S )
4		rngisoval.4	|- Y = ran J
5	1 2 3 4	isrngoiso	|- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngIso S ) <-> ( F e. ( R RngHom S ) /\ F : X -1-1-onto-> Y ) ) )
6	5	simplbda	|- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngIso S ) ) -> F : X -1-1-onto-> Y )
7	6	3impa	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngIso S ) ) -> F : X -1-1-onto-> Y )