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Theorem rngoisoval 30307
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 6304 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  RngHom  s )  =  ( R  RngHom  S ) )
2 fveq2 5872 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 rngisoval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2526 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5236 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 rngisoval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2526 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
8 f1oeq2 5814 . . . . 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 16 . . . 4  |-  ( r  =  R  ->  (
f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s )  <->  f : X
-1-1-onto-> ran  ( 1st `  s
) ) )
10 fveq2 5872 . . . . . . . 8  |-  ( s  =  S  ->  ( 1st `  s )  =  ( 1st `  S
) )
11 rngisoval.3 . . . . . . . 8  |-  J  =  ( 1st `  S
)
1210, 11syl6eqr 2526 . . . . . . 7  |-  ( s  =  S  ->  ( 1st `  s )  =  J )
1312rneqd 5236 . . . . . 6  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  ran  J )
14 rngisoval.4 . . . . . 6  |-  Y  =  ran  J
1513, 14syl6eqr 2526 . . . . 5  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  Y )
16 f1oeq3 5815 . . . . 5  |-  ( ran  ( 1st `  s
)  =  Y  -> 
( f : X -1-1-onto-> ran  ( 1st `  s )  <-> 
f : X -1-1-onto-> Y ) )
1715, 16syl 16 . . . 4  |-  ( s  =  S  ->  (
f : X -1-1-onto-> ran  ( 1st `  s )  <->  f : X
-1-1-onto-> Y ) )
189, 17sylan9bb 699 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s
)  <->  f : X -1-1-onto-> Y
) )
191, 18rabeqbidv 3113 . 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 30306 . 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 6320 . . 3  |-  ( R 
RngHom  S )  e.  _V
2221rabex 4604 . 2  |-  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }  e.  _V
2319, 20, 22ovmpt2a 6428 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   ran crn 5006   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   1stc1st 6793   RingOpscrngo 25200    RngHom crnghom 30290    RngIso crngiso 30291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-rngoiso 30306
This theorem is referenced by:  isrngoiso  30308
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