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Theorem elghomlem2 25187
Description: Lemma for elghom 25188. (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
elghomlem2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  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 ) ) ) ) )
Distinct variable groups:    x, f,
y, F    f, G, x, y    f, H, x, y
Allowed substitution hints:    S( x, y, f)

Proof of Theorem elghomlem2
StepHypRef Expression
1 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 ) ) ) }
21elghomlem1 25186 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
32eleq2d 2537 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  F  e.  S
) )
4 elex 3127 . . . . 5  |-  ( F  e.  S  ->  F  e.  _V )
5 feq1 5719 . . . . . . . 8  |-  ( f  =  F  ->  (
f : ran  G --> ran  H  <->  F : ran  G --> ran  H ) )
6 fveq1 5871 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
7 fveq1 5871 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
86, 7oveq12d 6313 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
) H ( f `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
9 fveq1 5871 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
108, 9eqeq12d 2489 . . . . . . . . 9  |-  ( f  =  F  ->  (
( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
11102ralbidv 2911 . . . . . . . 8  |-  ( f  =  F  ->  ( 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 ) ) ) )
125, 11anbi12d 710 . . . . . . 7  |-  ( f  =  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 : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1312, 1elab2g 3257 . . . . . 6  |-  ( F  e.  _V  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1413biimpd 207 . . . . 5  |-  ( F  e.  _V  ->  ( F  e.  S  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) ) )
154, 14mpcom 36 . . . 4  |-  ( F  e.  S  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
16 rnexg 6727 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
17 fex 6144 . . . . . . . 8  |-  ( ( F : ran  G --> ran  H  /\  ran  G  e.  _V )  ->  F  e.  _V )
1817expcom 435 . . . . . . 7  |-  ( ran 
G  e.  _V  ->  ( F : ran  G --> ran  H  ->  F  e.  _V ) )
1916, 18syl 16 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( F : ran  G --> ran  H  ->  F  e.  _V )
)
2019adantrd 468 . . . . 5  |-  ( G  e.  GrpOp  ->  ( ( 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  e.  _V ) )
2113biimprd 223 . . . . 5  |-  ( F  e.  _V  ->  (
( 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  e.  S )
)
2220, 21syli 37 . . . 4  |-  ( G  e.  GrpOp  ->  ( ( 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  e.  S
) )
2315, 22impbid2 204 . . 3  |-  ( G  e.  GrpOp  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
2423adantr 465 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  S  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
253, 24bitrd 253 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   _Vcvv 3118   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295   GrpOpcgr 25011   GrpOpHom cghom 25182
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-ghom 25183
This theorem is referenced by:  elghom  25188
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