Theorem reldmdprd 19906

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 19904	. 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 7475	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 395   = wceq 1541  {cab 2709  ∀wral 3047  {crab 3395   ∖ cdif 3894   ∩ cin 3896   ⊆ wss 3897  {csn 4571  ∪ cuni 4854   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5611  ran crn 5612   “ cima 5614  Rel wrel 5616  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341  Xcixp 8816   finSupp cfsupp 9240  0_gc0g 17338   Σ_g cgsu 17339  mrClscmrc 17480  Grpcgrp 18841  SubGrpcsubg 19028  Cntzccntz 19222   DProd cdprd 19902

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-dm 5621  df-oprab 7345  df-mpo 7346  df-dprd 19904

This theorem is referenced by:  dprddomprc  19909  dprdval0prc  19911  dprdval  19912  dprdgrp  19914  dprdf  19915  dprdssv  19925  subgdmdprd  19943  dprd2da  19951  dpjfval  19964