Theorem dprdf 20018

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 20009	. . . . . 6 ⊢ Rel dom DProd
2	1	brrelex2i 5741	. . . . 5 ⊢ (𝐺dom DProd 𝑆 → 𝑆 ∈ V)
3	2	dmexd 7921	. . . 4 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
4		eqid 2729	. . . 4 ⊢ dom 𝑆 = dom 𝑆
5		eqid 2729	. . . . 5 ⊢ (Cntz‘𝐺) = (Cntz‘𝐺)
6		eqid 2729	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		eqid 2729	. . . . 5 ⊢ (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
8	5, 6, 7	dmdprd 20010	. . . 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 584	. . 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 1140	1 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  ∀wral 3054  Vcvv 3472   ∖ cdif 3955   ∩ cin 3957   ⊆ wss 3958  {csn 4634  ∪ cuni 4916   class class class wbr 5154  dom cdm 5684   “ cima 5687  ⟶wf 6552  ‘cfv 6556  0_gc0g 17467  mrClscmrc 17609  Grpcgrp 18941  SubGrpcsubg 19128  Cntzccntz 19323   DProd cdprd 20005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5291  ax-sep 5305  ax-nul 5312  ax-pow 5371  ax-pr 5435  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3788  df-csb 3904  df-dif 3961  df-un 3963  df-in 3965  df-ss 3975  df-nul 4334  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-iun 5006  df-br 5155  df-opab 5217  df-mpt 5238  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-ixp 8931  df-dprd 20007

This theorem is referenced by:  dprdf2  20019  dprdsubg  20036  dprdspan  20039  subgdprd  20047  ablfaclem2  20098  ablfac2  20101