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Theorem rngoisoval 32260
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 6323 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  RngHom  s )  =  ( R  RngHom  S ) )
2 fveq2 5887 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 rngisoval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2513 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5080 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 rngisoval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2513 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
8 f1oeq2 5828 . . . . 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 17 . . . 4  |-  ( r  =  R  ->  (
f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s )  <->  f : X
-1-1-onto-> ran  ( 1st `  s
) ) )
10 fveq2 5887 . . . . . . . 8  |-  ( s  =  S  ->  ( 1st `  s )  =  ( 1st `  S
) )
11 rngisoval.3 . . . . . . . 8  |-  J  =  ( 1st `  S
)
1210, 11syl6eqr 2513 . . . . . . 7  |-  ( s  =  S  ->  ( 1st `  s )  =  J )
1312rneqd 5080 . . . . . 6  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  ran  J )
14 rngisoval.4 . . . . . 6  |-  Y  =  ran  J
1513, 14syl6eqr 2513 . . . . 5  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  Y )
16 f1oeq3 5829 . . . . 5  |-  ( ran  ( 1st `  s
)  =  Y  -> 
( f : X -1-1-onto-> ran  ( 1st `  s )  <-> 
f : X -1-1-onto-> Y ) )
1715, 16syl 17 . . . 4  |-  ( s  =  S  ->  (
f : X -1-1-onto-> ran  ( 1st `  s )  <->  f : X
-1-1-onto-> Y ) )
189, 17sylan9bb 711 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s
)  <->  f : X -1-1-onto-> Y
) )
191, 18rabeqbidv 3051 . 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 32259 . 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 6342 . . 3  |-  ( R 
RngHom  S )  e.  _V
2221rabex 4567 . 2  |-  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }  e.  _V
2319, 20, 22ovmpt2a 6453 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   {crab 2752   ran crn 4853   -1-1-onto->wf1o 5599   ` cfv 5600  (class class class)co 6314   1stc1st 6817   RingOpscrngo 26151    RngHom crnghom 32243    RngIso crngiso 32244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-rngoiso 32259
This theorem is referenced by:  isrngoiso  32261
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