Theorem reldmdprd 19122

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 19120	. 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 7288	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 398   = wceq 1536  {cab 2802  ∀wral 3141  {crab 3145   ∖ cdif 3936   ∩ cin 3938   ⊆ wss 3939  {csn 4570  ∪ cuni 4841   class class class wbr 5069   ↦ cmpt 5149  dom cdm 5558  ran crn 5559   “ cima 5561  Rel wrel 5563  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  Xcixp 8464   finSupp cfsupp 8836  0_gc0g 16716   Σ_g cgsu 16717  mrClscmrc 16857  Grpcgrp 18106  SubGrpcsubg 18276  Cntzccntz 18448   DProd cdprd 19118

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-dm 5568  df-oprab 7163  df-mpo 7164  df-dprd 19120

This theorem is referenced by:  dprddomprc  19125  dprdval0prc  19127  dprdval  19128  dprdgrp  19130  dprdf  19131  dprdssv  19141  subgdmdprd  19159  dprd2da  19167  dpjfval  19180