Theorem reldmdprd 20220

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 20218	. 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 7556	1 ⊢ Rel dom DProd

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  {cab 2738  ∀wral 3076  {crab 3412   ⊆ wss 3896   ∖ cdif 4044   ∩ cin 4088  {csn 4584  ∪ cuni 4866   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5650  ran crn 5651   “ cima 5653  Rel wrel 5655  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  Xcixp 8930   finSupp cfsupp 9358  0_gc0g 17617   Σ_g cgsu 17618  mrClscmrc 17760  Grpcgrp 19151  SubGrpcsubg 19337  Cntzccntz 19536   DProd cdprd 20216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-oprab 7426  df-mpo 7427  df-dprd 20218

This theorem is used by:  dprddomprc  20223  dprdval0prc  20225  dprdval  20226  dprdgrp  20228  dprdf  20229  dprdssv  20239  subgdmdprd  20257  dprd2da  20265  dpjfval  20278