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Theorem elghomlem2 24021
Description: Lemma for elghom 24022. (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 24020 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
32eleq2d 2524 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  F  e.  S
) )
4 elex 3087 . . . . 5  |-  ( F  e.  S  ->  F  e.  _V )
5 feq1 5653 . . . . . . . 8  |-  ( f  =  F  ->  (
f : ran  G --> ran  H  <->  F : ran  G --> ran  H ) )
6 fveq1 5801 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
7 fveq1 5801 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
86, 7oveq12d 6221 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
) H ( f `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
9 fveq1 5801 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
108, 9eqeq12d 2476 . . . . . . . . 9  |-  ( f  =  F  ->  (
( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
11102ralbidv 2879 . . . . . . . 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 3215 . . . . . 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 6623 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
17 fex 6062 . . . . . . . 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 1370    e. wcel 1758   {cab 2439   A.wral 2799   _Vcvv 3078   ran crn 4952   -->wf 5525   ` cfv 5529  (class class class)co 6203   GrpOpcgr 23845   GrpOpHom cghom 24016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-ghom 24017
This theorem is referenced by:  elghom  24022
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