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Theorem dprdf 18326
Description: The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdf (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))

Proof of Theorem dprdf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldmdprd 18317 . . . . . 6 Rel dom DProd
21brrelex2i 5119 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
3 dmexg 7044 . . . . 5 (𝑆 ∈ V → dom 𝑆 ∈ V)
42, 3syl 17 . . . 4 (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
5 eqid 2621 . . . 4 dom 𝑆 = dom 𝑆
6 eqid 2621 . . . . 5 (Cntz‘𝐺) = (Cntz‘𝐺)
7 eqid 2621 . . . . 5 (0g𝐺) = (0g𝐺)
8 eqid 2621 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
96, 7, 8dmdprd 18318 . . . 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})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
104, 5, 9sylancl 693 . . 3 (𝐺dom DProd 𝑆 → (𝐺dom DProd 𝑆 ↔ (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)}))))
1110ibi 256 . 2 (𝐺dom DProd 𝑆 → (𝐺 ∈ Grp ∧ 𝑆:dom 𝑆⟶(SubGrp‘𝐺) ∧ ∀𝑥 ∈ dom 𝑆(∀𝑦 ∈ (dom 𝑆 ∖ {𝑥})(𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)) ∧ ((𝑆𝑥) ∩ ((mrCls‘(SubGrp‘𝐺))‘ (𝑆 “ (dom 𝑆 ∖ {𝑥})))) = {(0g𝐺)})))
1211simp2d 1072 1 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  cin 3554  wss 3555  {csn 4148   cuni 4402   class class class wbr 4613  dom cdm 5074  cima 5077  wf 5843  cfv 5847  0gc0g 16021  mrClscmrc 16164  Grpcgrp 17343  SubGrpcsubg 17509  Cntzccntz 17669   DProd cdprd 18313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-ixp 7853  df-dprd 18315
This theorem is referenced by:  dprdf2  18327  dprdsubg  18344  dprdspan  18347  subgdprd  18355  ablfaclem2  18406  ablfac2  18409
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