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Theorem r0cld 21714
Description: The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
r0cld ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqffn 21701 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1125 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 Fn 𝑋)
4 fncnvima2 6490 . . . 4 (𝐹 Fn 𝑋 → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
53, 4syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}})
6 fvex 6350 . . . . . 6 (𝐹𝑧) ∈ V
76elsn 4324 . . . . 5 ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ (𝐹𝑧) = (𝐹𝐴))
8 simpl1 1204 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 simpr 479 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝑧𝑋)
10 simpl3 1208 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐴𝑋)
111kqfeq 21700 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
12 eleq2w 2811 . . . . . . . . 9 (𝑦 = 𝑜 → (𝑧𝑦𝑧𝑜))
13 eleq2w 2811 . . . . . . . . 9 (𝑦 = 𝑜 → (𝐴𝑦𝐴𝑜))
1412, 13bibi12d 334 . . . . . . . 8 (𝑦 = 𝑜 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑜𝐴𝑜)))
1514cbvralv 3298 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜))
1611, 15syl6bb 276 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
178, 9, 10, 16syl3anc 1463 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
187, 17syl5bb 272 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
1918rabbidva 3316 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}} = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
205, 19eqtrd 2782 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
211kqid 21704 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22213ad2ant1 1125 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23 simp2 1129 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ Fre)
24 simp3 1130 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐴𝑋)
25 fnfvelrn 6507 . . . . . 6 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
263, 24, 25syl2anc 696 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
271kqtopon 21703 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28273ad2ant1 1125 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29 toponuni 20892 . . . . . 6 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3028, 29syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → ran 𝐹 = (KQ‘𝐽))
3126, 30eleqtrd 2829 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ (KQ‘𝐽))
32 eqid 2748 . . . . 5 (KQ‘𝐽) = (KQ‘𝐽)
3332t1sncld 21303 . . . 4 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝐴) ∈ (KQ‘𝐽)) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
3423, 31, 33syl2anc 696 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35 cnclima 21245 . . 3 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3622, 34, 35syl2anc 696 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3720, 36eqeltrrd 2828 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)} ∈ (Clsd‘𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  {crab 3042  {csn 4309   cuni 4576  cmpt 4869  ccnv 5253  ran crn 5255  cima 5257   Fn wfn 6032  cfv 6037  (class class class)co 6801  TopOnctopon 20888  Clsdccld 20993   Cn ccn 21201  Frect1 21284  KQckq 21669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-map 8013  df-qtop 16340  df-top 20872  df-topon 20889  df-cld 20996  df-cn 21204  df-t1 21291  df-kq 21670
This theorem is referenced by:  nrmr0reg  21725
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