Metamath Proof Explorer


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 R
rngisoval.2 X = ran G
rngisoval.3 J = 1 st S
rngisoval.4 Y = 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 R
2 rngisoval.2 X = ran G
3 rngisoval.3 J = 1 st S
4 rngisoval.4 Y = 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 f : X 1-1 onto Y F : X 1-1 onto Y
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 |-