Theorem reldmghm 13693

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 13692	. 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 6080	1 |- Rel dom GrpHom

Syntax hints:   /\ wa 104   = wceq 1373   {cab 2193   A.wral 2486   [.wsbc 3005   dom cdm 4693   Rel wrel 4698   -->wf 5286   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   Grpcgrp 13447   GrpHom cghm 13691

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-dm 4703  df-oprab 5971  df-mpo 5972  df-ghm 13692

This theorem is referenced by: (None)