Theorem reldmghm 13312

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 13311	. 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 6030	1 |- Rel dom GrpHom

Syntax hints:   /\ wa 104   = wceq 1364   {cab 2179   A.wral 2472   [.wsbc 2985   dom cdm 4659   Rel wrel 4664   -->wf 5250   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   GrpHom cghm 13310

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-dm 4669  df-oprab 5922  df-mpo 5923  df-ghm 13311

This theorem is referenced by: (None)