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Theorem elghomlem2 21942
Description: Lemma for elghom 21943. (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 21941 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( G GrpOpHom  H )  =  S )
32eleq2d 2502 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  F  e.  S
) )
4 elex 2956 . . . . 5  |-  ( F  e.  S  ->  F  e.  _V )
5 feq1 5568 . . . . . . . 8  |-  ( f  =  F  ->  (
f : ran  G --> ran  H  <->  F : ran  G --> ran  H ) )
6 fveq1 5719 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
7 fveq1 5719 . . . . . . . . . . 11  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
86, 7oveq12d 6091 . . . . . . . . . 10  |-  ( f  =  F  ->  (
( f `  x
) H ( f `
 y ) )  =  ( ( F `
 x ) H ( F `  y
) ) )
9 fveq1 5719 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  ( x G y ) )  =  ( F `  ( x G y ) ) )
108, 9eqeq12d 2449 . . . . . . . . 9  |-  ( f  =  F  ->  (
( ( f `  x ) H ( f `  y ) )  =  ( f `
 ( x G y ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) )
11102ralbidv 2739 . . . . . . . 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 692 . . . . . . 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 3076 . . . . . 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 199 . . . . 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 34 . . . 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 5123 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
17 fex 5961 . . . . . . . 8  |-  ( ( F : ran  G --> ran  H  /\  ran  G  e.  _V )  ->  F  e.  _V )
1817expcom 425 . . . . . . 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 455 . . . . 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 215 . . . . 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 35 . . . 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 196 . . 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 452 . 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 245 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   _Vcvv 2948   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   GrpOpcgr 21766   GrpOpHom cghom 21937
This theorem is referenced by:  elghom  21943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-ghom 21938
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