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Theorem elghomlem1OLD 25564
Description: Obsolete as of 15-Mar-2020. Lemma for elghomOLD 25566. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
elghomlem1OLD.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
elghomlem1OLD  |-  ( ( 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 elghomlem1OLD
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 6705 . . 3  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
2 rnexg 6705 . . 3  |-  ( H  e.  GrpOp  ->  ran  H  e. 
_V )
3 elghomlem1OLD.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 6729 . . 3  |-  ( ( ran  G  e.  _V  /\ 
ran  H  e.  _V )  ->  S  e.  _V )
51, 2, 4syl2an 475 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  S  e.  _V )
6 rneq 5217 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
76feq2d 5700 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g --> ran  h  <->  f : ran  G --> ran  h ) )
8 oveq 6276 . . . . . . . . 9  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
98fveq2d 5852 . . . . . . . 8  |-  ( g  =  G  ->  (
f `  ( x
g y ) )  =  ( f `  ( x G y ) ) )
109eqeq2d 2468 . . . . . . 7  |-  ( g  =  G  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x g y ) )  <->  ( (
f `  x )
h ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
116, 10raleqbidv 3065 . . . . . 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 3065 . . . . 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 708 . . . 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 2590 . . 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 5217 . . . . . . 7  |-  ( h  =  H  ->  ran  h  =  ran  H )
1615feq3d 5701 . . . . . 6  |-  ( h  =  H  ->  (
f : ran  G --> ran  h  <->  f : ran  G --> ran  H ) )
17 oveq 6276 . . . . . . . 8  |-  ( h  =  H  ->  (
( f `  x
) h ( f `
 y ) )  =  ( ( f `
 x ) H ( f `  y
) ) )
1817eqeq1d 2456 . . . . . . 7  |-  ( h  =  H  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( (
f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
19182ralbidv 2898 . . . . . 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 ) ) ) )
2016, 19anbi12d 708 . . . . 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 ) ) ) ) )
2120abbidv 2590 . . . 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 ) ) ) } )
2221, 3syl6eqr 2513 . . 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 )
23 df-ghomOLD 25561 . . 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 ) ) ) } )
2414, 22, 23ovmpt2g 6410 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  S  e.  _V )  ->  ( G GrpOpHom  H )  =  S )
255, 24mpd3an3 1323 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   _Vcvv 3106   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   GrpOpcgr 25389   GrpOpHom cghomOLD 25560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-ghomOLD 25561
This theorem is referenced by:  elghomlem2OLD  25565
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