Theorem reldmdprd 20030

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 20028	. 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 7525	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 399   = wceq 1559  {cab 2739  ∀wral 3075  {crab 3413   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  {csn 4579  ∪ cuni 4862   class class class wbr 5097   ↦ cmpt 5178  dom cdm 5643  ran crn 5644   “ cima 5646  Rel wrel 5648  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391  Xcixp 8873   finSupp cfsupp 9301  0_gc0g 17459   Σ_g cgsu 17460  mrClscmrc 17602  Grpcgrp 18966  SubGrpcsubg 19153  Cntzccntz 19346   DProd cdprd 20026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-dm 5653  df-oprab 7395  df-mpo 7396  df-dprd 20028

This theorem is referenced by:  dprddomprc  20033  dprdval0prc  20035  dprdval  20036  dprdgrp  20038  dprdf  20039  dprdssv  20049  subgdmdprd  20067  dprd2da  20075  dpjfval  20088