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Theorem rngoisoval 32123
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 6258 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  RngHom  s )  =  ( R  RngHom  S ) )
2 fveq2 5825 . . . . . . . 8  |-  ( r  =  R  ->  ( 1st `  r )  =  ( 1st `  R
) )
3 rngisoval.1 . . . . . . . 8  |-  G  =  ( 1st `  R
)
42, 3syl6eqr 2480 . . . . . . 7  |-  ( r  =  R  ->  ( 1st `  r )  =  G )
54rneqd 5024 . . . . . 6  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  ran  G )
6 rngisoval.2 . . . . . 6  |-  X  =  ran  G
75, 6syl6eqr 2480 . . . . 5  |-  ( r  =  R  ->  ran  ( 1st `  r )  =  X )
8 f1oeq2 5766 . . . . 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 5825 . . . . . . . 8  |-  ( s  =  S  ->  ( 1st `  s )  =  ( 1st `  S
) )
11 rngisoval.3 . . . . . . . 8  |-  J  =  ( 1st `  S
)
1210, 11syl6eqr 2480 . . . . . . 7  |-  ( s  =  S  ->  ( 1st `  s )  =  J )
1312rneqd 5024 . . . . . 6  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  ran  J )
14 rngisoval.4 . . . . . 6  |-  Y  =  ran  J
1513, 14syl6eqr 2480 . . . . 5  |-  ( s  =  S  ->  ran  ( 1st `  s )  =  Y )
16 f1oeq3 5767 . . . . 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 704 . . 3  |-  ( ( r  =  R  /\  s  =  S )  ->  ( f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s
)  <->  f : X -1-1-onto-> Y
) )
191, 18rabeqbidv 3017 . 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 32122 . 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 6277 . . 3  |-  ( R 
RngHom  S )  e.  _V
2221rabex 4518 . 2  |-  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y }  e.  _V
2319, 20, 22ovmpt2a 6385 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 187    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2718   ran crn 4797   -1-1-onto->wf1o 5543   ` cfv 5544  (class class class)co 6249   1stc1st 6749   RingOpscrngo 26045    RngHom crnghom 32106    RngIso crngiso 32107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-rngoiso 32122
This theorem is referenced by:  isrngoiso  32124
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