Theorem reldmghm 13909

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.

Step	Hyp	Ref	Expression
1		df-ghm 13908	. 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 ) ) ) } )
2	1	reldmmpo 6143	1 |- Rel dom GrpHom

Syntax hints:   /\ wa 104   = wceq 1398   {cab 2217   A.wral 2511   [.wsbc 3032   dom cdm 4731   Rel wrel 4736   -->wf 5329   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   Grpcgrp 13663   GrpHom cghm 13907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-dm 4741  df-oprab 6032  df-mpo 6033  df-ghm 13908

This theorem is referenced by: (None)