Theorem reldmdprd 19919

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 19917	. 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 7489	1 ⊢ Rel dom DProd

Syntax hints:   ∧ wa 395   = wceq 1541  {cab 2711  ∀wral 3048  {crab 3396   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  {csn 4577  ∪ cuni 4860   class class class wbr 5095   ↦ cmpt 5176  dom cdm 5621  ran crn 5622   “ cima 5624  Rel wrel 5626  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  Xcixp 8831   finSupp cfsupp 9256  0_gc0g 17350   Σ_g cgsu 17351  mrClscmrc 17493  Grpcgrp 18854  SubGrpcsubg 19041  Cntzccntz 19235   DProd cdprd 19915

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-dm 5631  df-oprab 7359  df-mpo 7360  df-dprd 19917

This theorem is referenced by:  dprddomprc  19922  dprdval0prc  19924  dprdval  19925  dprdgrp  19927  dprdf  19928  dprdssv  19938  subgdmdprd  19956  dprd2da  19964  dpjfval  19977