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Theorem reldmdprd 18604
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 18602 . 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 6918 1 Rel dom DProd
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  {cab 2757  wral 3061  {crab 3065  cdif 3720  cin 3722  wss 3723  {csn 4316   cuni 4574   class class class wbr 4786  cmpt 4863  dom cdm 5249  ran crn 5250  cima 5252  Rel wrel 5254  wf 6027  cfv 6031  (class class class)co 6793  Xcixp 8062   finSupp cfsupp 8431  0gc0g 16308   Σg cgsu 16309  mrClscmrc 16451  Grpcgrp 17630  SubGrpcsubg 17796  Cntzccntz 17955   DProd cdprd 18600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-dm 5259  df-oprab 6797  df-mpt2 6798  df-dprd 18602
This theorem is referenced by:  dprddomprc  18607  dprdval0prc  18609  dprdval  18610  dprdgrp  18612  dprdf  18613  dprdssv  18623  subgdmdprd  18641  dprd2da  18649  dpjfval  18662
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