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Theorem elghomlem1 21081
Description: Lemma for elghom 21083. (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 4977 . . 3  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
2 rnexg 4977 . . 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 5460 . . 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 4941 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
76feq2d 5417 . . . . 5  |-  ( g  =  G  ->  (
f : ran  g --> ran  h  <->  f : ran  G --> ran  h ) )
8 oveq 5906 . . . . . . . . 9  |-  ( g  =  G  ->  (
x g y )  =  ( x G y ) )
98fveq2d 5567 . . . . . . . 8  |-  ( g  =  G  ->  (
f `  ( x
g y ) )  =  ( f `  ( x G y ) ) )
109eqeq2d 2327 . . . . . . 7  |-  ( g  =  G  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x g y ) )  <->  ( (
f `  x )
h ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
116, 10raleqbidv 2782 . . . . . 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 2782 . . . . 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 2430 . . 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 4941 . . . . . . 7  |-  ( h  =  H  ->  ran  h  =  ran  H )
16 feq3 5414 . . . . . . 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 5906 . . . . . . . 8  |-  ( h  =  H  ->  (
( f `  x
) h ( f `
 y ) )  =  ( ( f `
 x ) H ( f `  y
) ) )
1918eqeq1d 2324 . . . . . . 7  |-  ( h  =  H  ->  (
( ( f `  x ) h ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( (
f `  x ) H ( f `  y ) )  =  ( f `  (
x G y ) ) ) )
20192ralbidv 2619 . . . . . 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 2430 . . . 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 2366 . . 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 21078 . . 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 6024 . 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 1633    e. wcel 1701   {cab 2302   A.wral 2577   _Vcvv 2822   ran crn 4727   -->wf 5288   ` cfv 5292  (class class class)co 5900   GrpOpcgr 20906   GrpOpHom cghom 21077
This theorem is referenced by:  elghomlem2  21082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-ghom 21078
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