Theorem reldmdprd 20032

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 20030	. 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 7567	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 395   = wceq 1537  {cab 2712  ∀wral 3059  {crab 3433   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  {csn 4631  ∪ cuni 4912   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689  ran crn 5690   “ cima 5692  Rel wrel 5694  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Xcixp 8936   finSupp cfsupp 9399  0_gc0g 17486   Σ_g cgsu 17487  mrClscmrc 17628  Grpcgrp 18964  SubGrpcsubg 19151  Cntzccntz 19346   DProd cdprd 20028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-oprab 7435  df-mpo 7436  df-dprd 20030

This theorem is referenced by:  dprddomprc  20035  dprdval0prc  20037  dprdval  20038  dprdgrp  20040  dprdf  20041  dprdssv  20051  subgdmdprd  20069  dprd2da  20077  dpjfval  20090