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Theorem rngoisoval 26020
Description: The set of ring isomorphisms. (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
rngoisoval  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }
)
Distinct variable groups:    R, f    S, f    f, X    f, Y
Allowed substitution hints:    G( f)    J( f)

Proof of Theorem rngoisoval
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 5867 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  RngHom  s )  =  ( R  RngHom  S ) )
2 fveq2 5525 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 rngisoval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2333 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 4906 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 rngisoval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2333 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
8 f1oeq2 5464 . . . . 5  |-  ( ran  ( 1st `  r
)  =  X  -> 
( f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s
)  <->  f : X -1-1-onto-> ran  ( 1st `  s ) ) )
97, 8syl 15 . . . 4  |-  ( r  =  R  ->  (
f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s )  <->  f : X
-1-1-onto-> ran  ( 1st `  s
) ) )
10 fveq2 5525 . . . . . . . 8  |-  ( s  =  S  ->  ( 1st `  s )  =  ( 1st `  S
) )
11 rngisoval.3 . . . . . . . 8  |-  J  =  ( 1st `  S
)
1210, 11syl6eqr 2333 . . . . . . 7  |-  ( s  =  S  ->  ( 1st `  s )  =  J )
1312rneqd 4906 . . . . . 6  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  ran  J )
14 rngisoval.4 . . . . . 6  |-  Y  =  ran  J
1513, 14syl6eqr 2333 . . . . 5  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  Y )
16 f1oeq3 5465 . . . . 5  |-  ( ran  ( 1st `  s
)  =  Y  -> 
( f : X -1-1-onto-> ran  ( 1st `  s )  <-> 
f : X -1-1-onto-> Y ) )
1715, 16syl 15 . . . 4  |-  ( s  =  S  ->  (
f : X -1-1-onto-> ran  ( 1st `  s )  <->  f : X
-1-1-onto-> Y ) )
189, 17sylan9bb 680 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s
)  <->  f : X -1-1-onto-> Y
) )
191, 18rabeqbidv 2783 . 2  |-  ( ( r  =  R  /\  s  =  S )  ->  { f  e.  ( r  RngHom  s )  |  f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s ) }  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y } )
20 df-rngoiso 26019 . 2  |-  RngIso  =  ( r  e.  RingOps ,  s  e.  RingOps  |->  { f  e.  ( r  RngHom  s )  |  f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s
) } )
21 ovex 5883 . . 3  |-  ( R 
RngHom  S )  e.  _V
2221rabex 4165 . 2  |-  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }  e.  _V
2319, 20, 22ovmpt2a 5978 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   ran crn 4690   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1stc1st 6120   RingOpscrngo 21042    RngHom crnghom 26003    RngIso crngiso 26004
This theorem is referenced by:  isrngoiso  26021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-rngoiso 26019
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