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Theorem reldmghm 19140
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.
StepHypRef Expression
1 df-ghm 19139 . 2 GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔[(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥𝑤𝑦𝑤 (𝑔‘(𝑥(+g𝑠)𝑦)) = ((𝑔𝑥)(+g𝑡)(𝑔𝑦)))})
21reldmmpo 7539 1 Rel dom GrpHom
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  {cab 2703  wral 3055  [wsbc 3772  dom cdm 5669  Rel wrel 5674  wf 6533  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Grpcgrp 18863   GrpHom cghm 19138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-dm 5679  df-oprab 7409  df-mpo 7410  df-ghm 19139
This theorem is referenced by:  ghmquskerco  33035
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