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Theorem dislly 23400
Description: The discrete space 𝒫 𝑋 is locally 𝐴 if and only if every singleton space has property 𝐴. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
dislly (𝑋𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉   𝑥,𝑋

Proof of Theorem dislly
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 768 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝒫 𝑋 ∈ Locally 𝐴)
2 simpr 484 . . . . . 6 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
3 vex 3475 . . . . . . 7 𝑥 ∈ V
43snelpw 5447 . . . . . 6 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
52, 4sylib 217 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
6 vsnid 4666 . . . . . 6 𝑥 ∈ {𝑥}
76a1i 11 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
8 llyi 23377 . . . . 5 ((𝒫 𝑋 ∈ Locally 𝐴 ∧ {𝑥} ∈ 𝒫 𝑋𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
91, 5, 7, 8syl3anc 1369 . . . 4 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
10 simpr1 1192 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11 simpr2 1193 . . . . . . . . . . 11 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑥𝑦)
1211snssd 4813 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
1310, 12eqssd 3997 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
1413oveq2d 7436 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = (𝒫 𝑋t {𝑥}))
15 simplll 774 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑋𝑉)
16 simplr 768 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑥𝑋)
1716snssd 4813 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18 restdis 23081 . . . . . . . . 9 ((𝑋𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
1915, 17, 18syl2anc 583 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
2014, 19eqtrd 2768 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = 𝒫 {𝑥})
21 simpr3 1194 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) ∈ 𝐴)
2220, 21eqeltrrd 2830 . . . . . 6 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
2322ex 412 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ((𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
2423rexlimdvw 3157 . . . 4 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → (∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
259, 24mpd 15 . . 3 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝒫 {𝑥} ∈ 𝐴)
2625ralrimiva 3143 . 2 ((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴)
27 distop 22897 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2827adantr 480 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29 elpwi 4610 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3029adantl 481 . . . . . . . 8 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → 𝑦𝑋)
31 ssralv 4048 . . . . . . . 8 (𝑦𝑋 → (∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥𝑦 𝒫 {𝑥} ∈ 𝐴))
3230, 31syl 17 . . . . . . 7 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → (∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥𝑦 𝒫 {𝑥} ∈ 𝐴))
33 simprl 770 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥𝑦)
3433snssd 4813 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
3530adantr 480 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑦𝑋)
3634, 35sstrd 3990 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37 vsnex 5431 . . . . . . . . . . . . 13 {𝑥} ∈ V
3837elpw 4607 . . . . . . . . . . . 12 ({𝑥} ∈ 𝒫 𝑋 ↔ {𝑥} ⊆ 𝑋)
3936, 38sylibr 233 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑋)
4037elpw 4607 . . . . . . . . . . . 12 ({𝑥} ∈ 𝒫 𝑦 ↔ {𝑥} ⊆ 𝑦)
4134, 40sylibr 233 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑦)
4239, 41elind 4194 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43 snidg 4663 . . . . . . . . . . 11 (𝑥𝑦𝑥 ∈ {𝑥})
4443ad2antrl 727 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45 simpll 766 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋𝑉)
4645, 36, 18syl2anc 583 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
47 simprr 772 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
4846, 47eqeltrd 2829 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) ∈ 𝐴)
49 eleq2 2818 . . . . . . . . . . . 12 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
50 oveq2 7428 . . . . . . . . . . . . 13 (𝑢 = {𝑥} → (𝒫 𝑋t 𝑢) = (𝒫 𝑋t {𝑥}))
5150eleq1d 2814 . . . . . . . . . . . 12 (𝑢 = {𝑥} → ((𝒫 𝑋t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋t {𝑥}) ∈ 𝐴))
5249, 51anbi12d 631 . . . . . . . . . . 11 (𝑢 = {𝑥} → ((𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋t {𝑥}) ∈ 𝐴)))
5352rspcev 3609 . . . . . . . . . 10 (({𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦) ∧ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋t {𝑥}) ∈ 𝐴)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5442, 44, 48, 53syl12anc 836 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5554expr 456 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ 𝑥𝑦) → (𝒫 {𝑥} ∈ 𝐴 → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
5655ralimdva 3164 . . . . . . 7 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → (∀𝑥𝑦 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
5732, 56syld 47 . . . . . 6 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → (∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴 → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
5857imp 406 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5958an32s 651 . . . 4 (((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) ∧ 𝑦 ∈ 𝒫 𝑋) → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
6059ralrimiva 3143 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
61 islly 23371 . . 3 (𝒫 𝑋 ∈ Locally 𝐴 ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
6228, 60, 61sylanbrc 582 . 2 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
6326, 62impbida 800 1 (𝑋𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3058  wrex 3067  cin 3946  wss 3947  𝒫 cpw 4603  {csn 4629  (class class class)co 7420  t crest 17401  Topctop 22794  Locally clly 23367
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 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-en 8964  df-fin 8967  df-fi 9434  df-rest 17403  df-topgen 17424  df-top 22795  df-topon 22812  df-bases 22848  df-lly 23369
This theorem is referenced by:  disllycmp  23401  dis1stc  23402
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