MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldmghm Unicode version

Theorem reldmghm 14698
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 14697 . 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 5971 1  |-  Rel  dom  GrpHom
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   {cab 2282   A.wral 2556   [.wsbc 3004   dom cdm 4705   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378    GrpHom cghm 14696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-oprab 5878  df-mpt2 5879  df-ghm 14697
  Copyright terms: Public domain W3C validator