Theorem reldmdprd 19928

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 19926	. 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 7492	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 395   = wceq 1541  {cab 2714  ∀wral 3051  {crab 3399   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  {csn 4580  ∪ cuni 4863   class class class wbr 5098   ↦ cmpt 5179  dom cdm 5624  ran crn 5625   “ cima 5627  Rel wrel 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Xcixp 8835   finSupp cfsupp 9264  0_gc0g 17359   Σ_g cgsu 17360  mrClscmrc 17502  Grpcgrp 18863  SubGrpcsubg 19050  Cntzccntz 19244   DProd cdprd 19924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7362  df-mpo 7363  df-dprd 19926

This theorem is referenced by:  dprddomprc  19931  dprdval0prc  19933  dprdval  19934  dprdgrp  19936  dprdf  19937  dprdssv  19947  subgdmdprd  19965  dprd2da  19973  dpjfval  19986