Theorem reldmdprd 19965

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 19963	. 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 7494	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 395   = wceq 1542  {cab 2715  ∀wral 3052  {crab 3390   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  {csn 4568  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5624  ran crn 5625   “ cima 5627  Rel wrel 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  Xcixp 8838   finSupp cfsupp 9267  0_gc0g 17393   Σ_g cgsu 17394  mrClscmrc 17536  Grpcgrp 18900  SubGrpcsubg 19087  Cntzccntz 19281   DProd cdprd 19961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7364  df-mpo 7365  df-dprd 19963

This theorem is referenced by:  dprddomprc  19968  dprdval0prc  19970  dprdval  19971  dprdgrp  19973  dprdf  19974  dprdssv  19984  subgdmdprd  20002  dprd2da  20010  dpjfval  20023