Theorem dprdf 19590

Description: The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)

Assertion

Ref	Expression
dprdf	⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))

Proof of Theorem dprdf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldmdprd 19581	. . . . . 6 ⊢ Rel dom DProd
2	1	brrelex2i 5643	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
3	2	dmexd 7739	. . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
4		eqid 2739	. . . 4 ⊢ dom 𝑆 = dom 𝑆
5		eqid 2739	. . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
6		eqid 2739	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		eqid 2739	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
8	5, 6, 7	dmdprd 19582	. . . 4 ⊢ ((dom 𝑆 ∈ V ∧ dom 𝑆 = dom 𝑆) → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)}))))
9	3, 4, 8	sylancl 585	. . 3 ⊢ (𝐺dom DProd 𝑆 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)}))))
10	9	ibi 266	. 2 ⊢ (𝐺dom DProd 𝑆 → (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆‘𝑦)) ∧ ((𝑆‘𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘∪ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g‘𝐺)})))
11	10	simp2d 1141	1 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  Vcvv 3430   ∖ cdif 3888   ∩ cin 3890   ⊆ wss 3891  {csn 4566  ∪ cuni 4844   class class class wbr 5078  dom cdm 5588   “ cima 5591  ⟶wf 6426  ‘cfv 6430  0_gc0g 17131  mrClscmrc 17273  Grpcgrp 18558  SubGrpcsubg 18730  Cntzccntz 18902   DProd cdprd 19577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-ixp 8660  df-dprd 19579

This theorem is referenced by:  dprdf2  19591  dprdsubg  19608  dprdspan  19611  subgdprd  19619  ablfaclem2  19670  ablfac2  19673