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 ℎ 𝑓 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dprd 20015	. 2 ⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓)))
2	1	reldmmpo 7567	1 ⊢ Rel dom DProd

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  0_gc0g 17484   Σ_g 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