Theorem reldmdprd 19121

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 19119	. 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 7287	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 398   = wceq 1537  {cab 2801  ∀wral 3140  {crab 3144   ∖ cdif 3935   ∩ cin 3937   ⊆ wss 3938  {csn 4569  ∪ cuni 4840   class class class wbr 5068   ↦ cmpt 5148  dom cdm 5557  ran crn 5558   “ cima 5560  Rel wrel 5562  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Xcixp 8463   finSupp cfsupp 8835  0_gc0g 16715   Σ_g cgsu 16716  mrClscmrc 16856  Grpcgrp 18105  SubGrpcsubg 18275  Cntzccntz 18447   DProd cdprd 19117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-dm 5567  df-oprab 7162  df-mpo 7163  df-dprd 19119

This theorem is referenced by:  dprddomprc  19124  dprdval0prc  19126  dprdval  19127  dprdgrp  19129  dprdf  19130  dprdssv  19140  subgdmdprd  19158  dprd2da  19166  dpjfval  19179