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Theorem rngoisoval 26711
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 5883 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  RngHom  s )  =  ( R  RngHom  S ) )
2 fveq2 5541 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 rngisoval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2346 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 4922 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 rngisoval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2346 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
8 f1oeq2 5480 . . . . 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 5541 . . . . . . . 8  |-  ( s  =  S  ->  ( 1st `  s )  =  ( 1st `  S
) )
11 rngisoval.3 . . . . . . . 8  |-  J  =  ( 1st `  S
)
1210, 11syl6eqr 2346 . . . . . . 7  |-  ( s  =  S  ->  ( 1st `  s )  =  J )
1312rneqd 4922 . . . . . 6  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  ran  J )
14 rngisoval.4 . . . . . 6  |-  Y  =  ran  J
1513, 14syl6eqr 2346 . . . . 5  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  Y )
16 f1oeq3 5481 . . . . 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 2796 . 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 26710 . 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 5899 . . 3  |-  ( R 
RngHom  S )  e.  _V
2221rabex 4181 . 2  |-  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }  e.  _V
2319, 20, 22ovmpt2a 5994 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 1632    e. wcel 1696   {crab 2560   ran crn 4706   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1stc1st 6136   RingOpscrngo 21058    RngHom crnghom 26694    RngIso crngiso 26695
This theorem is referenced by:  isrngoiso  26712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-rngoiso 26710
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