Theorem reldmdprd 20065

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 20063	. 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 7542	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 400   = wceq 1567  {cab 2747  ∀wral 3085  {crab 3423   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913  {csn 4591  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5659  ran crn 5660   “ cima 5662  Rel wrel 5664  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  Xcixp 8891   finSupp cfsupp 9317  0_gc0g 17488   Σ_g cgsu 17489  mrClscmrc 17631  Grpcgrp 18996  SubGrpcsubg 19182  Cntzccntz 19381   DProd cdprd 20061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-dm 5669  df-oprab 7412  df-mpo 7413  df-dprd 20063

This theorem is referenced by:  dprddomprc  20068  dprdval0prc  20070  dprdval  20071  dprdgrp  20073  dprdf  20074  dprdssv  20084  subgdmdprd  20102  dprd2da  20110  dpjfval  20123