Theorem reldmdprd 19936

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 19934	. 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 7526	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 395   = wceq 1540  {cab 2708  ∀wral 3045  {crab 3408   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  {csn 4592  ∪ cuni 4874   class class class wbr 5110   ↦ cmpt 5191  dom cdm 5641  ran crn 5642   “ cima 5644  Rel wrel 5646  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Xcixp 8873   finSupp cfsupp 9319  0_gc0g 17409   Σ_g cgsu 17410  mrClscmrc 17551  Grpcgrp 18872  SubGrpcsubg 19059  Cntzccntz 19254   DProd cdprd 19932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-dm 5651  df-oprab 7394  df-mpo 7395  df-dprd 19934

This theorem is referenced by:  dprddomprc  19939  dprdval0prc  19941  dprdval  19942  dprdgrp  19944  dprdf  19945  dprdssv  19955  subgdmdprd  19973  dprd2da  19981  dpjfval  19994