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Theorem dislly 23001
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 486 . . . . . 6 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
3 vex 3479 . . . . . . 7 𝑥 ∈ V
43snelpw 5446 . . . . . 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 22978 . . . . 5 ((𝒫 𝑋 ∈ Locally 𝐴 ∧ {𝑥} ∈ 𝒫 𝑋𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
91, 5, 7, 8syl3anc 1372 . . . 4 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
10 simpr1 1195 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11 simpr2 1196 . . . . . . . . . . 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 4000 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
1413oveq2d 7425 . . . . . . . 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 22682 . . . . . . . . 9 ((𝑋𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
1915, 17, 18syl2anc 585 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
2014, 19eqtrd 2773 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = 𝒫 {𝑥})
21 simpr3 1197 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) ∈ 𝐴)
2220, 21eqeltrrd 2835 . . . . . 6 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
2322ex 414 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ((𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
2423rexlimdvw 3161 . . . 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 3147 . 2 ((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴)
27 distop 22498 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2827adantr 482 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29 elpwi 4610 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3029adantl 483 . . . . . . . 8 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → 𝑦𝑋)
31 ssralv 4051 . . . . . . . 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 482 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑦𝑋)
3634, 35sstrd 3993 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37 vsnex 5430 . . . . . . . . . . . . 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 4195 . . . . . . . . . 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 585 . . . . . . . . . . 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 2834 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) ∈ 𝐴)
49 eleq2 2823 . . . . . . . . . . . 12 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
50 oveq2 7417 . . . . . . . . . . . . 13 (𝑢 = {𝑥} → (𝒫 𝑋t 𝑢) = (𝒫 𝑋t {𝑥}))
5150eleq1d 2819 . . . . . . . . . . . 12 (𝑢 = {𝑥} → ((𝒫 𝑋t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋t {𝑥}) ∈ 𝐴))
5249, 51anbi12d 632 . . . . . . . . . . 11 (𝑢 = {𝑥} → ((𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋t {𝑥}) ∈ 𝐴)))
5352rspcev 3613 . . . . . . . . . 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 458 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ 𝑥𝑦) → (𝒫 {𝑥} ∈ 𝐴 → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
5655ralimdva 3168 . . . . . . 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 408 . . . . 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 3147 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
61 islly 22972 . . 3 (𝒫 𝑋 ∈ Locally 𝐴 ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
6228, 60, 61sylanbrc 584 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cin 3948  wss 3949  𝒫 cpw 4603  {csn 4629  (class class class)co 7409  t crest 17366  Topctop 22395  Locally clly 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-en 8940  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-lly 22970
This theorem is referenced by:  disllycmp  23002  dis1stc  23003
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