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Theorem reldmdprd 18377
Description: The domain of the internal direct product operation is a relation. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.)
Assertion
Ref Expression
reldmdprd Rel dom DProd

Proof of Theorem reldmdprd
Dummy variables 𝑔 𝑓 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dprd 18375 . 2 DProd = (𝑔 ∈ Grp, 𝑠 ∈ { ∣ (:dom ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom (∀𝑦 ∈ (dom ∖ {𝑥})(𝑥) ⊆ ((Cntz‘𝑔)‘(𝑦)) ∧ ((𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘ ( “ (dom ∖ {𝑥})))) = {(0g𝑔)}))} ↦ ran (𝑓 ∈ {X𝑥 ∈ dom 𝑠(𝑠𝑥) ∣ finSupp (0g𝑔)} ↦ (𝑔 Σg 𝑓)))
21reldmmpt2 6756 1 Rel dom DProd
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1481  {cab 2606  wral 2909  {crab 2913  cdif 3564  cin 3566  wss 3567  {csn 4168   cuni 4427   class class class wbr 4644  cmpt 4720  dom cdm 5104  ran crn 5105  cima 5107  Rel wrel 5109  wf 5872  cfv 5876  (class class class)co 6635  Xcixp 7893   finSupp cfsupp 8260  0gc0g 16081   Σg cgsu 16082  mrClscmrc 16224  Grpcgrp 17403  SubGrpcsubg 17569  Cntzccntz 17729   DProd cdprd 18373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-dm 5114  df-oprab 6639  df-mpt2 6640  df-dprd 18375
This theorem is referenced by:  dprddomprc  18380  dprdval0prc  18382  dprdval  18383  dprdgrp  18385  dprdf  18386  dprdssv  18396  subgdmdprd  18414  dprd2da  18422  dpjfval  18435
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