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Theorem elghom 24022
Description: Membership in the set of group homomorphisms from  G to  H. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
elghom.1  |-  X  =  ran  G
elghom.2  |-  W  =  ran  H
Assertion
Ref Expression
elghom  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> W  /\  A. x  e.  X  A. y  e.  X  ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) ) ) )
Distinct variable groups:    x, F, y    x, G, y    x, H, y    x, X, y
Allowed substitution hints:    W( x, y)

Proof of Theorem elghom
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eqid 2454 . . 3  |-  { 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 ) ) ) }
21elghomlem2 24021 . 2  |-  ( ( 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 ) ) ) ) )
3 elghom.1 . . . 4  |-  X  =  ran  G
4 elghom.2 . . . 4  |-  W  =  ran  H
53, 4feq23i 5664 . . 3  |-  ( F : X --> W  <->  F : ran  G --> ran  H )
63raleqi 3027 . . . 4  |-  ( A. y  e.  X  (
( F `  x
) H ( F `
 y ) )  =  ( F `  ( x G y ) )  <->  A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `  (
x G y ) ) )
73, 6raleqbii 2848 . . 3  |-  ( A. x  e.  X  A. y  e.  X  (
( 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 ) ) )
85, 7anbi12i 697 . 2  |-  ( ( F : X --> W  /\  A. x  e.  X  A. y  e.  X  (
( 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 ) ) ) )
92, 8syl6bbr 263 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : X
--> W  /\  A. x  e.  X  A. y  e.  X  ( ( 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   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:  ghomlin  24023  ghomid  24024  ghomgrpilem1  27468  ghomgrpilem2  27469  ghomsn  27471  ghomfo  27474  ghomgsg  27476  ghomf  28915  ghomco  28916  rngogrphom  28945
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