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Theorem isrngoiso 26286
Description: The predicate "is a ring isomorphism between  R and  S." (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
rngisoval.1  |-  G  =  ( 1st `  R
)
rngisoval.2  |-  X  =  ran  G
rngisoval.3  |-  J  =  ( 1st `  S
)
rngisoval.4  |-  Y  =  ran  J
Assertion
Ref Expression
isrngoiso  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S )  /\  F : X -1-1-onto-> Y
) ) )

Proof of Theorem isrngoiso
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 rngisoval.1 . . . 4  |-  G  =  ( 1st `  R
)
2 rngisoval.2 . . . 4  |-  X  =  ran  G
3 rngisoval.3 . . . 4  |-  J  =  ( 1st `  S
)
4 rngisoval.4 . . . 4  |-  Y  =  ran  J
51, 2, 3, 4rngoisoval 26285 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }
)
65eleq2d 2455 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  F  e.  { f  e.  ( R 
RngHom  S )  |  f : X -1-1-onto-> Y } ) )
7 f1oeq1 5606 . . 3  |-  ( f  =  F  ->  (
f : X -1-1-onto-> Y  <->  F : X
-1-1-onto-> Y ) )
87elrab 3036 . 2  |-  ( F  e.  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y } 
<->  ( F  e.  ( R  RngHom  S )  /\  F : X -1-1-onto-> Y ) )
96, 8syl6bb 253 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S )  /\  F : X -1-1-onto-> Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2654   ran crn 4820   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   1stc1st 6287   RingOpscrngo 21812    RngHom crnghom 26268    RngIso crngiso 26269
This theorem is referenced by:  rngoiso1o  26287  rngoisohom  26288  rngoisocnv  26289  rngoisoco  26290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-rngoiso 26284
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