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Theorem isrngoiso 28931
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 28930 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }
)
65eleq2d 2524 . 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 5739 . . 3  |-  ( f  =  F  ->  (
f : X -1-1-onto-> Y  <->  F : X
-1-1-onto-> Y ) )
87elrab 3222 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2802   ran crn 4948   -1-1-onto->wf1o 5524   ` cfv 5525  (class class class)co 6199   1stc1st 6684   RingOpscrngo 24013    RngHom crnghom 28913    RngIso crngiso 28914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-rngoiso 28929
This theorem is referenced by:  rngoiso1o  28932  rngoisohom  28933  rngoisocnv  28934  rngoisoco  28935
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