Theorem reldmghm 13372

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 13371	. 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 6034	1 |- Rel dom GrpHom

Syntax hints:   /\ wa 104   = wceq 1364   {cab 2182   A.wral 2475   [.wsbc 2989   dom cdm 4663   Rel wrel 4668   -->wf 5254   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   Grpcgrp 13132   GrpHom cghm 13370

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-dm 4673  df-oprab 5926  df-mpo 5927  df-ghm 13371

This theorem is referenced by: (None)