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Theorem reldmghm 15005
Description: Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
reldmghm  |-  Rel  dom  GrpHom

Proof of Theorem reldmghm
Dummy variables  g 
s  t  w  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ghm 15004 . 2  |-  GrpHom  =  ( s  e.  Grp , 
t  e.  Grp  |->  { g  |  [. ( Base `  s )  /  w ]. ( g : w --> ( Base `  t
)  /\  A. x  e.  w  A. y  e.  w  ( g `  ( x ( +g  `  s ) y ) )  =  ( ( g `  x ) ( +g  `  t
) ( g `  y ) ) ) } )
21reldmmpt2 6181 1  |-  Rel  dom  GrpHom
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   {cab 2422   A.wral 2705   [.wsbc 3161   dom cdm 4878   Rel wrel 4883   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685    GrpHom cghm 15003
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-dm 4888  df-oprab 6085  df-mpt2 6086  df-ghm 15004
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