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Theorem elghomlem1 25067
Description: Lemma for elghom 25069. (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 6716 . . 3  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
2 rnexg 6716 . . 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 6740 . . 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 5228 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
76feq2d 5718 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g --> ran  h  <->  f : ran  G --> ran  h ) )
8 oveq 6290 . . . . . . . . 9  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
98fveq2d 5870 . . . . . . . 8  |-  ( g  =  G  ->  (
f `  ( x
g y ) )  =  ( f `  ( x G y ) ) )
109eqeq2d 2481 . . . . . . 7  |-  ( g  =  G  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x g y ) )  <->  ( (
f `  x )
h ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
116, 10raleqbidv 3072 . . . . . 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 3072 . . . . 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 2603 . . 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 5228 . . . . . . 7  |-  ( h  =  H  ->  ran  h  =  ran  H )
16 feq3 5715 . . . . . . 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 6290 . . . . . . . 8  |-  ( h  =  H  ->  (
( f `  x
) h ( f `
 y ) )  =  ( ( f `
 x ) H ( f `  y
) ) )
1918eqeq1d 2469 . . . . . . 7  |-  ( h  =  H  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( (
f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
20192ralbidv 2908 . . . . . 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 2603 . . . 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 2526 . . 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 25064 . . 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 6421 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  S  e.  _V )  ->  ( G GrpOpHom  H )  =  S )
265, 25mpd3an3 1325 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 1379    e. wcel 1767   {cab 2452   A.wral 2814   _Vcvv 3113   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   GrpOpcgr 24892   GrpOpHom cghom 25063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-ghom 25064
This theorem is referenced by:  elghomlem2  25068
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