Theorem reldmghm 13611

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 13610	. 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 6059	1 |- Rel dom GrpHom

Syntax hints:   /\ wa 104   = wceq 1373   {cab 2191   A.wral 2484   [.wsbc 2998   dom cdm 4676   Rel wrel 4681   -->wf 5268   ` cfv 5272  (class class class)co 5946   Basecbs 12865   +g cplusg 12942   Grpcgrp 13365   GrpHom cghm 13609

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-dm 4686  df-oprab 5950  df-mpo 5951  df-ghm 13610

This theorem is referenced by: (None)