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Theorem reldmdprd 20017
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 20015 . 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)))
21reldmmpo 7567 1 Rel dom DProd
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  {cab 2714  wral 3061  {crab 3436  cdif 3948  cin 3950  wss 3951  {csn 4626   cuni 4907   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  cima 5688  Rel wrel 5690  wf 6557  cfv 6561  (class class class)co 7431  Xcixp 8937   finSupp cfsupp 9401  0gc0g 17484   Σg cgsu 17485  mrClscmrc 17626  Grpcgrp 18951  SubGrpcsubg 19138  Cntzccntz 19333   DProd cdprd 20013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-oprab 7435  df-mpo 7436  df-dprd 20015
This theorem is referenced by:  dprddomprc  20020  dprdval0prc  20022  dprdval  20023  dprdgrp  20025  dprdf  20026  dprdssv  20036  subgdmdprd  20054  dprd2da  20062  dpjfval  20075
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