Theorem reldmghm 13980

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 13979	. 2 ⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∣ [(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑔‘(𝑥(+g‘𝑠)𝑦)) = ((𝑔‘𝑥)(+g‘𝑡)(𝑔‘𝑦)))})
2	1	reldmmpo 6167	1 ⊢ Rel dom GrpHom

Syntax hints:   ∧ wa 104   = wceq 1398  {cab 2220  ∀wral 2522  [wsbc 3044  dom cdm 4751  Rel wrel 4756  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052  Basecbs 13233  +_gcplusg 13311  Grpcgrp 13734   GrpHom cghm 13978

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 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-rel 4758  df-dm 4761  df-oprab 6056  df-mpo 6057  df-ghm 13979

This theorem is referenced by: (None)