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Theorem elghomlem1 21028
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
elghomlem1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
Distinct variable groups:    x, f,
y, G    f, H, x, y
Allowed substitution hints:    S( x, y, f)

Proof of Theorem elghomlem1
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 4940 . . 3  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
2 rnexg 4940 . . 3  |-  ( H  e.  GrpOp  ->  ran  H  e. 
_V )
3 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 ) ) ) }
43fabexg 5422 . . 3  |-  ( ( ran  G  e.  _V  /\ 
ran  H  e.  _V )  ->  S  e.  _V )
51, 2, 4syl2an 463 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  S  e.  _V )
6 rneq 4904 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
76feq2d 5380 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g --> ran  h  <->  f : ran  G --> ran  h ) )
8 oveq 5864 . . . . . . . . 9  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
98fveq2d 5529 . . . . . . . 8  |-  ( g  =  G  ->  (
f `  ( x
g y ) )  =  ( f `  ( x G y ) ) )
109eqeq2d 2294 . . . . . . 7  |-  ( g  =  G  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x g y ) )  <->  ( (
f `  x )
h ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
116, 10raleqbidv 2748 . . . . . 6  |-  ( g  =  G  ->  ( A. y  e.  ran  g ( ( f `
 x ) h ( f `  y
) )  =  ( f `  ( x g y ) )  <->  A. y  e.  ran  G ( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) ) ) )
126, 11raleqbidv 2748 . . . . 5  |-  ( g  =  G  ->  ( 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 ) ) ) )
137, 12anbi12d 691 . . . 4  |-  ( g  =  G  ->  (
( 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 ) ) ) ) )
1413abbidv 2397 . . 3  |-  ( g  =  G  ->  { 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  |  ( f : ran  G --> ran  h  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) ) ) } )
15 rneq 4904 . . . . . . 7  |-  ( h  =  H  ->  ran  h  =  ran  H )
16 feq3 5377 . . . . . . 7  |-  ( ran  h  =  ran  H  ->  ( f : ran  G --> ran  h  <->  f : ran  G --> ran  H )
)
1715, 16syl 15 . . . . . 6  |-  ( h  =  H  ->  (
f : ran  G --> ran  h  <->  f : ran  G --> ran  H ) )
18 oveq 5864 . . . . . . . 8  |-  ( h  =  H  ->  (
( f `  x
) h ( f `
 y ) )  =  ( ( f `
 x ) H ( f `  y
) ) )
1918eqeq1d 2291 . . . . . . 7  |-  ( h  =  H  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( (
f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
20192ralbidv 2585 . . . . . 6  |-  ( h  =  H  ->  ( 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 ) ) ) )
2117, 20anbi12d 691 . . . . 5  |-  ( h  =  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 ) ) )  <->  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) ) ) ) )
2221abbidv 2397 . . . 4  |-  ( h  =  H  ->  { 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  |  ( f : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) ) ) } )
2322, 3syl6eqr 2333 . . 3  |-  ( h  =  H  ->  { 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 ) ) ) }  =  S )
24 df-ghom 21025 . . 3  |- GrpOpHom  =  ( g  e.  GrpOp ,  h  e.  GrpOp  |->  { 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 ) ) ) } )
2514, 23, 24ovmpt2g 5982 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  S  e.  _V )  ->  ( G GrpOpHom  H )  =  S )
265, 25mpd3an3 1278 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
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:  elghomlem2  21029
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-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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ghom 21025
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