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Theorem reldmghm 16869
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 16868 . 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 6417 1  |-  Rel  dom  GrpHom
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   {cab 2407   A.wral 2775   [.wsbc 3299   dom cdm 4849   Rel wrel 4854   -->wf 5593   ` cfv 5597  (class class class)co 6301   Basecbs 15108   +g cplusg 15177   Grpcgrp 16656    GrpHom cghm 16867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-xp 4855  df-rel 4856  df-dm 4859  df-oprab 6305  df-mpt2 6306  df-ghm 16868
This theorem is referenced by: (None)
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