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Theorem isrngoiso 30587
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 30586 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }
)
65eleq2d 2466 . 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 5732 . . 3  |-  ( f  =  F  ->  (
f : X -1-1-onto-> Y  <->  F : X
-1-1-onto-> Y ) )
87elrab 3199 . 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 367    = wceq 1399    e. wcel 1836   {crab 2750   ran crn 4931   -1-1-onto->wf1o 5512   ` cfv 5513  (class class class)co 6218   1stc1st 6719   RingOpscrngo 25519    RngHom crnghom 30569    RngIso crngiso 30570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-rngoiso 30585
This theorem is referenced by:  rngoiso1o  30588  rngoisohom  30589  rngoisocnv  30590  rngoisoco  30591
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