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Theorem dislly 21210
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 791 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝒫 𝑋 ∈ Locally 𝐴)
2 simpr 477 . . . . . 6 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
3 vex 3189 . . . . . . 7 𝑥 ∈ V
43snelpw 4874 . . . . . 6 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
52, 4sylib 208 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
6 vsnid 4180 . . . . . 6 𝑥 ∈ {𝑥}
76a1i 11 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
8 llyi 21187 . . . . 5 ((𝒫 𝑋 ∈ Locally 𝐴 ∧ {𝑥} ∈ 𝒫 𝑋𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
91, 5, 7, 8syl3anc 1323 . . . 4 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
10 simpr1 1065 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11 simpr2 1066 . . . . . . . . . . 11 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑥𝑦)
1211snssd 4309 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
1310, 12eqssd 3600 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
1413oveq2d 6620 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = (𝒫 𝑋t {𝑥}))
15 simplll 797 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑋𝑉)
16 simplr 791 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑥𝑋)
1716snssd 4309 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18 restdis 20892 . . . . . . . . 9 ((𝑋𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
1915, 17, 18syl2anc 692 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
2014, 19eqtrd 2655 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = 𝒫 {𝑥})
21 simpr3 1067 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) ∈ 𝐴)
2220, 21eqeltrrd 2699 . . . . . 6 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
2322ex 450 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ((𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
2423rexlimdvw 3027 . . . 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 2960 . 2 ((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴)
27 distop 20710 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2827adantr 481 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29 elpwi 4140 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3029adantl 482 . . . . . . . 8 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → 𝑦𝑋)
31 ssralv 3645 . . . . . . . 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 793 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥𝑦)
3433snssd 4309 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
3530adantr 481 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑦𝑋)
3634, 35sstrd 3593 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37 snex 4869 . . . . . . . . . . . . 13 {𝑥} ∈ V
3837elpw 4136 . . . . . . . . . . . 12 ({𝑥} ∈ 𝒫 𝑋 ↔ {𝑥} ⊆ 𝑋)
3936, 38sylibr 224 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑋)
4037elpw 4136 . . . . . . . . . . . 12 ({𝑥} ∈ 𝒫 𝑦 ↔ {𝑥} ⊆ 𝑦)
4134, 40sylibr 224 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑦)
4239, 41elind 3776 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43 snidg 4177 . . . . . . . . . . 11 (𝑥𝑦𝑥 ∈ {𝑥})
4443ad2antrl 763 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45 simpll 789 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋𝑉)
4645, 36, 18syl2anc 692 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
47 simprr 795 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
4846, 47eqeltrd 2698 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) ∈ 𝐴)
49 eleq2 2687 . . . . . . . . . . . 12 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
50 oveq2 6612 . . . . . . . . . . . . 13 (𝑢 = {𝑥} → (𝒫 𝑋t 𝑢) = (𝒫 𝑋t {𝑥}))
5150eleq1d 2683 . . . . . . . . . . . 12 (𝑢 = {𝑥} → ((𝒫 𝑋t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋t {𝑥}) ∈ 𝐴))
5249, 51anbi12d 746 . . . . . . . . . . 11 (𝑢 = {𝑥} → ((𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋t {𝑥}) ∈ 𝐴)))
5352rspcev 3295 . . . . . . . . . 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 1321 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5554expr 642 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ 𝑥𝑦) → (𝒫 {𝑥} ∈ 𝐴 → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
5655ralimdva 2956 . . . . . . 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 445 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5958an32s 845 . . . 4 (((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) ∧ 𝑦 ∈ 𝒫 𝑋) → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
6059ralrimiva 2960 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
61 islly 21181 . . 3 (𝒫 𝑋 ∈ Locally 𝐴 ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
6228, 60, 61sylanbrc 697 . 2 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
6326, 62impbida 876 1 (𝑋𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148  (class class class)co 6604  t crest 16002  Topctop 20617  Locally clly 21177
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-3or 1037  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-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  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-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  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-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-lly 21179
This theorem is referenced by:  disllycmp  21211  dis1stc  21212
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