Theorem reldmdprd 19859

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 19857	. 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 7538	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 397   = wceq 1542  {cab 2710  ∀wral 3062  {crab 3433   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  {csn 4627  ∪ cuni 4907   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675  ran crn 5676   “ cima 5678  Rel wrel 5680  ⟶wf 6536  ‘cfv 6540  (class class class)co 7404  Xcixp 8887   finSupp cfsupp 9357  0_gc0g 17381   Σ_g cgsu 17382  mrClscmrc 17523  Grpcgrp 18815  SubGrpcsubg 18994  Cntzccntz 19173   DProd cdprd 19855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-dm 5685  df-oprab 7408  df-mpo 7409  df-dprd 19857

This theorem is referenced by:  dprddomprc  19862  dprdval0prc  19864  dprdval  19865  dprdgrp  19867  dprdf  19868  dprdssv  19878  subgdmdprd  19896  dprd2da  19904  dpjfval  19917