Theorem isrngoiso

Description: The predicate "is a ring isomorphism between R and S ". (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Hypotheses	rngisoval.1	$$⊢G=1^{st}\left(R\right)$$
	rngisoval.2	$$⊢X=\mathrm{ran}G$$
	rngisoval.3	$$⊢J=1^{st}\left(S\right)$$
	rngisoval.4	$$⊢Y=\mathrm{ran}J$$
Assertion	isrngoiso	Could not format assertion : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> ( F e. ( R RingOpsHom S ) /\ F : X -1-1-onto-> Y ) ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		rngisoval.1	$$⊢G=1^{st}\left(R\right)$$
2		rngisoval.2	$$⊢X=\mathrm{ran}G$$
3		rngisoval.3	$$⊢J=1^{st}\left(S\right)$$
4		rngisoval.4	$$⊢Y=\mathrm{ran}J$$
5	1 2 3 4	rngoisoval	Could not format ( ( R e. RingOps /\ S e. RingOps ) -> ( R RingOpsIso S ) = { f e. ( R RingOpsHom S ) | f : X -1-1-onto-> Y } ) : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps ) -> ( R RingOpsIso S ) = { f e. ( R RingOpsHom S ) | f : X -1-1-onto-> Y } ) with typecode |-
6	5	eleq2d	Could not format ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> F e. { f e. ( R RingOpsHom S ) | f : X -1-1-onto-> Y } ) ) : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> F e. { f e. ( R RingOpsHom S ) | f : X -1-1-onto-> Y } ) ) with typecode |-
7		f1oeq1	$$⊢f=F\to \left(f:X\underset{1-1\; onto}{⟶}Y\leftrightarrow F:X\underset{1-1\; onto}{⟶}Y\right)$$
8	7	elrab	Could not format ( F e. { f e. ( R RingOpsHom S ) | f : X -1-1-onto-> Y } <-> ( F e. ( R RingOpsHom S ) /\ F : X -1-1-onto-> Y ) ) : No typesetting found for |- ( F e. { f e. ( R RingOpsHom S ) | f : X -1-1-onto-> Y } <-> ( F e. ( R RingOpsHom S ) /\ F : X -1-1-onto-> Y ) ) with typecode |-
9	6 8	bitrdi	Could not format ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> ( F e. ( R RingOpsHom S ) /\ F : X -1-1-onto-> Y ) ) ) : No typesetting found for |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RingOpsIso S ) <-> ( F e. ( R RingOpsHom S ) /\ F : X -1-1-onto-> Y ) ) ) with typecode |-