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Theorem dprdf 18605
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 18596 . . . . . 6 Rel dom DProd
21brrelex2i 5316 . . . . 5 (𝐺dom DProd 𝑆𝑆 ∈ V)
3 dmexg 7262 . . . . 5 (𝑆 ∈ V → dom 𝑆 ∈ V)
42, 3syl 17 . . . 4 (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
5 eqid 2760 . . . 4 dom 𝑆 = dom 𝑆
6 eqid 2760 . . . . 5 (Cntz‘𝐺) = (Cntz‘𝐺)
7 eqid 2760 . . . . 5 (0g𝐺) = (0g𝐺)
8 eqid 2760 . . . . 5 (mrCls‘(SubGrp‘𝐺)) = (mrCls‘(SubGrp‘𝐺))
96, 7, 8dmdprd 18597 . . . 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 697 . . 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 1138 1 (𝐺dom DProd 𝑆𝑆:dom 𝑆⟶(SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cdif 3712  cin 3714  wss 3715  {csn 4321   cuni 4588   class class class wbr 4804  dom cdm 5266  cima 5269  wf 6045  cfv 6049  0gc0g 16302  mrClscmrc 16445  Grpcgrp 17623  SubGrpcsubg 17789  Cntzccntz 17948   DProd cdprd 18592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ixp 8075  df-dprd 18594
This theorem is referenced by:  dprdf2  18606  dprdsubg  18623  dprdspan  18626  subgdprd  18634  ablfaclem2  18685  ablfac2  18688
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