Theorem reldmdprd 20093

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 20091	. 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 7550	1 ⊢ Rel dom DProd

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  {cab 2743  ∀wral 3081  {crab 3418   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ∪ cuni 4874   class class class wbr 5111   ↦ cmpt 5194  dom cdm 5663  ran crn 5664   “ cima 5666  Rel wrel 5668  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  Xcixp 8897   finSupp cfsupp 9324  0_gc0g 17510   Σ_g cgsu 17511  mrClscmrc 17653  Grpcgrp 19024  SubGrpcsubg 19210  Cntzccntz 19409   DProd cdprd 20089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7420  df-mpo 7421  df-dprd 20091

This theorem is used by:  dprddomprc  20096  dprdval0prc  20098  dprdval  20099  dprdgrp  20101  dprdf  20102  dprdssv  20112  subgdmdprd  20130  dprd2da  20138  dpjfval  20151