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Theorem r0cld 21464
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 21451 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
323ad2ant1 1080 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 Fn 𝑋)
4 fncnvima2 6300 . . . 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 6163 . . . . . 6 (𝐹𝑧) ∈ V
76elsn 4168 . . . . 5 ((𝐹𝑧) ∈ {(𝐹𝐴)} ↔ (𝐹𝑧) = (𝐹𝐴))
8 simpl1 1062 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9 simpr 477 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝑧𝑋)
10 simpl3 1064 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) ∧ 𝑧𝑋) → 𝐴𝑋)
111kqfeq 21450 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑦𝐽 (𝑧𝑦𝐴𝑦)))
12 eleq2 2687 . . . . . . . . 9 (𝑦 = 𝑜 → (𝑧𝑦𝑧𝑜))
13 eleq2 2687 . . . . . . . . 9 (𝑦 = 𝑜 → (𝐴𝑦𝐴𝑜))
1412, 13bibi12d 335 . . . . . . . 8 (𝑦 = 𝑜 → ((𝑧𝑦𝐴𝑦) ↔ (𝑧𝑜𝐴𝑜)))
1514cbvralv 3162 . . . . . . 7 (∀𝑦𝐽 (𝑧𝑦𝐴𝑦) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜))
1611, 15syl6bb 276 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝐴𝑋) → ((𝐹𝑧) = (𝐹𝐴) ↔ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)))
178, 9, 10, 16syl3anc 1323 . . . . 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 3179 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {𝑧𝑋 ∣ (𝐹𝑧) ∈ {(𝐹𝐴)}} = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
205, 19eqtrd 2655 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) = {𝑧𝑋 ∣ ∀𝑜𝐽 (𝑧𝑜𝐴𝑜)})
211kqid 21454 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22213ad2ant1 1080 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23 simp2 1060 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ Fre)
24 simp3 1061 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → 𝐴𝑋)
25 fnfvelrn 6317 . . . . . 6 ((𝐹 Fn 𝑋𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
263, 24, 25syl2anc 692 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ ran 𝐹)
271kqtopon 21453 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28273ad2ant1 1080 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29 toponuni 20651 . . . . . 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 2700 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹𝐴) ∈ (KQ‘𝐽))
32 eqid 2621 . . . . 5 (KQ‘𝐽) = (KQ‘𝐽)
3332t1sncld 21053 . . . 4 (((KQ‘𝐽) ∈ Fre ∧ (𝐹𝐴) ∈ (KQ‘𝐽)) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
3423, 31, 33syl2anc 692 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35 cnclima 20995 . . 3 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3622, 34, 35syl2anc 692 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴𝑋) → (𝐹 “ {(𝐹𝐴)}) ∈ (Clsd‘𝐽))
3720, 36eqeltrrd 2699 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  {csn 4153   cuni 4407  cmpt 4678  ccnv 5078  ran crn 5080  cima 5082   Fn wfn 5847  cfv 5852  (class class class)co 6610  TopOnctopon 20647  Clsdccld 20743   Cn ccn 20951  Frect1 21034  KQckq 21419
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-qtop 16099  df-top 20631  df-topon 20648  df-cld 20746  df-cn 20954  df-t1 21041  df-kq 21420
This theorem is referenced by:  nrmr0reg  21475
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