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Theorem reldmdprd 19113
 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 19111 . 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 7279 1 Rel dom DProd
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 398   = wceq 1533  {cab 2799  ∀wral 3138  {crab 3142   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  {csn 4561  ∪ cuni 4832   class class class wbr 5059   ↦ cmpt 5139  dom cdm 5550  ran crn 5551   “ cima 5553  Rel wrel 5555  ⟶wf 6346  ‘cfv 6350  (class class class)co 7150  Xcixp 8455   finSupp cfsupp 8827  0gc0g 16707   Σg cgsu 16708  mrClscmrc 16848  Grpcgrp 18097  SubGrpcsubg 18267  Cntzccntz 18439   DProd cdprd 19109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-rel 5557  df-dm 5560  df-oprab 7154  df-mpo 7155  df-dprd 19111 This theorem is referenced by:  dprddomprc  19116  dprdval0prc  19118  dprdval  19119  dprdgrp  19121  dprdf  19122  dprdssv  19132  subgdmdprd  19150  dprd2da  19158  dpjfval  19171
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