Theorem reldmdprd 19241

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 19239	. 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 7303	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 399   = wceq 1542  {cab 2717  ∀wral 3054  {crab 3058   ∖ cdif 3841   ∩ cin 3843   ⊆ wss 3844  {csn 4517  ∪ cuni 4797   class class class wbr 5031   ↦ cmpt 5111  dom cdm 5526  ran crn 5527   “ cima 5529  Rel wrel 5531  ⟶wf 6336  ‘cfv 6340  (class class class)co 7173  Xcixp 8510   finSupp cfsupp 8909  0_gc0g 16819   Σ_g cgsu 16820  mrClscmrc 16960  Grpcgrp 18222  SubGrpcsubg 18394  Cntzccntz 18566   DProd cdprd 19237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-br 5032  df-opab 5094  df-xp 5532  df-rel 5533  df-dm 5536  df-oprab 7177  df-mpo 7178  df-dprd 19239

This theorem is referenced by:  dprddomprc  19244  dprdval0prc  19246  dprdval  19247  dprdgrp  19249  dprdf  19250  dprdssv  19260  subgdmdprd  19278  dprd2da  19286  dpjfval  19299