Theorem risci

Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Assertion	risci	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RingOpsIso S ) ) -> R ~=R S )

Proof

Step	Hyp	Ref	Expression
1		elex2	|- ( F e. ( R RingOpsIso S ) -> E. f f e. ( R RingOpsIso S ) )
2		risc	|- ( ( R e. RingOps /\ S e. RingOps ) -> ( R ~=R S <-> E. f f e. ( R RingOpsIso S ) ) )
3	1 2	imbitrrid	|- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) -> R ~=R S ) )
4	3	3impia	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RingOpsIso S ) ) -> R ~=R S )