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Theorem elghomlem1 23853
Description: Lemma for elghom 23855. (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 6515 . . 3  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
2 rnexg 6515 . . 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 6538 . . 3  |-  ( ( ran  G  e.  _V  /\ 
ran  H  e.  _V )  ->  S  e.  _V )
51, 2, 4syl2an 477 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  S  e.  _V )
6 rneq 5070 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
76feq2d 5552 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g --> ran  h  <->  f : ran  G --> ran  h ) )
8 oveq 6102 . . . . . . . . 9  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
98fveq2d 5700 . . . . . . . 8  |-  ( g  =  G  ->  (
f `  ( x
g y ) )  =  ( f `  ( x G y ) ) )
109eqeq2d 2454 . . . . . . 7  |-  ( g  =  G  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x g y ) )  <->  ( (
f `  x )
h ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
116, 10raleqbidv 2936 . . . . . 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 2936 . . . . 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 710 . . . 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 2562 . . 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 5070 . . . . . . 7  |-  ( h  =  H  ->  ran  h  =  ran  H )
16 feq3 5549 . . . . . . 7  |-  ( ran  h  =  ran  H  ->  ( f : ran  G --> ran  h  <->  f : ran  G --> ran  H )
)
1715, 16syl 16 . . . . . 6  |-  ( h  =  H  ->  (
f : ran  G --> ran  h  <->  f : ran  G --> ran  H ) )
18 oveq 6102 . . . . . . . 8  |-  ( h  =  H  ->  (
( f `  x
) h ( f `
 y ) )  =  ( ( f `
 x ) H ( f `  y
) ) )
1918eqeq1d 2451 . . . . . . 7  |-  ( h  =  H  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( (
f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
20192ralbidv 2762 . . . . . 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 710 . . . . 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 2562 . . . 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 2493 . . 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 23850 . . 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 6230 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  S  e.  _V )  ->  ( G GrpOpHom  H )  =  S )
265, 25mpd3an3 1315 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   _Vcvv 2977   ran crn 4846   -->wf 5419   ` cfv 5423  (class class class)co 6096   GrpOpcgr 23678   GrpOpHom cghom 23849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-ghom 23850
This theorem is referenced by:  elghomlem2  23854
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