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Theorem elghomlem2 21029
Description: Lemma for elghom 21030. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
elghomlem1.1  |-  S  =  { f  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) }
Assertion
Ref Expression
elghomlem2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
Distinct variable groups:    x, f,
y, F    f, G, x, y    f, H, x, y
Allowed substitution hints:    S( x, y, f)

Proof of Theorem elghomlem2
StepHypRef Expression
1 elghomlem1.1 . . . 4  |-  S  =  { f  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) }
21elghomlem1 21028 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
32eleq2d 2350 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  F  e.  S
) )
4 elex 2796 . . . . 5  |-  ( F  e.  S  ->  F  e.  _V )
5 feq1 5375 . . . . . . . 8  |-  ( f  =  F  ->  (
f : ran  G --> ran  H  <->  F : ran  G --> ran  H ) )
6 fveq1 5524 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
7 fveq1 5524 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
86, 7oveq12d 5876 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
) H ( f `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
9 fveq1 5524 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
108, 9eqeq12d 2297 . . . . . . . . 9  |-  ( f  =  F  ->  (
( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
11102ralbidv 2585 . . . . . . . 8  |-  ( f  =  F  ->  ( A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
125, 11anbi12d 691 . . . . . . 7  |-  ( f  =  F  ->  (
( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x
) H ( f `
 y ) )  =  ( f `  ( x G y ) ) )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1312, 1elab2g 2916 . . . . . 6  |-  ( F  e.  _V  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1413biimpd 198 . . . . 5  |-  ( F  e.  _V  ->  ( F  e.  S  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) ) )
154, 14mpcom 32 . . . 4  |-  ( F  e.  S  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
16 rnexg 4940 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
17 fex 5749 . . . . . . . 8  |-  ( ( F : ran  G --> ran  H  /\  ran  G  e.  _V )  ->  F  e.  _V )
1817expcom 424 . . . . . . 7  |-  ( ran 
G  e.  _V  ->  ( F : ran  G --> ran  H  ->  F  e.  _V ) )
1916, 18syl 15 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( F : ran  G --> ran  H  ->  F  e.  _V )
)
2019adantrd 454 . . . . 5  |-  ( G  e.  GrpOp  ->  ( ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  _V ) )
2113biimprd 214 . . . . 5  |-  ( F  e.  _V  ->  (
( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) ) )  ->  F  e.  S )
)
2220, 21syli 33 . . . 4  |-  ( G  e.  GrpOp  ->  ( ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) )  ->  F  e.  S
) )
2315, 22impbid2 195 . . 3  |-  ( G  e.  GrpOp  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
2423adantr 451 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
253, 24bitrd 244 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   _Vcvv 2788   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   GrpOpHom cghom 21024
This theorem is referenced by:  elghom  21030
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  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-ghom 21025
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