MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dislly Structured version   Visualization version   GIF version

Theorem dislly 21789
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 765 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝒫 𝑋 ∈ Locally 𝐴)
2 simpr 485 . . . . . 6 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
3 vex 3440 . . . . . . 7 𝑥 ∈ V
43snelpw 5229 . . . . . 6 (𝑥𝑋 ↔ {𝑥} ∈ 𝒫 𝑋)
52, 4sylib 219 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → {𝑥} ∈ 𝒫 𝑋)
6 vsnid 4507 . . . . . 6 𝑥 ∈ {𝑥}
76a1i 11 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → 𝑥 ∈ {𝑥})
8 llyi 21766 . . . . 5 ((𝒫 𝑋 ∈ Locally 𝐴 ∧ {𝑥} ∈ 𝒫 𝑋𝑥 ∈ {𝑥}) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
91, 5, 7, 8syl3anc 1364 . . . 4 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ∃𝑦 ∈ 𝒫 𝑋(𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴))
10 simpr1 1187 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 ⊆ {𝑥})
11 simpr2 1188 . . . . . . . . . . 11 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑥𝑦)
1211snssd 4649 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
1310, 12eqssd 3906 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑦 = {𝑥})
1413oveq2d 7032 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = (𝒫 𝑋t {𝑥}))
15 simplll 771 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑋𝑉)
16 simplr 765 . . . . . . . . . 10 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝑥𝑋)
1716snssd 4649 . . . . . . . . 9 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
18 restdis 21470 . . . . . . . . 9 ((𝑋𝑉 ∧ {𝑥} ⊆ 𝑋) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
1915, 17, 18syl2anc 584 . . . . . . . 8 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
2014, 19eqtrd 2831 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) = 𝒫 {𝑥})
21 simpr3 1189 . . . . . . 7 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → (𝒫 𝑋t 𝑦) ∈ 𝐴)
2220, 21eqeltrrd 2884 . . . . . 6 ((((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) ∧ (𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
2322ex 413 . . . . 5 (((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) ∧ 𝑥𝑋) → ((𝑦 ⊆ {𝑥} ∧ 𝑥𝑦 ∧ (𝒫 𝑋t 𝑦) ∈ 𝐴) → 𝒫 {𝑥} ∈ 𝐴))
2423rexlimdvw 3253 . . . 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 3149 . 2 ((𝑋𝑉 ∧ 𝒫 𝑋 ∈ Locally 𝐴) → ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴)
27 distop 21287 . . . 4 (𝑋𝑉 → 𝒫 𝑋 ∈ Top)
2827adantr 481 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Top)
29 elpwi 4463 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3029adantl 482 . . . . . . . 8 ((𝑋𝑉𝑦 ∈ 𝒫 𝑋) → 𝑦𝑋)
31 ssralv 3954 . . . . . . . 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 767 . . . . . . . . . . . . . 14 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥𝑦)
3433snssd 4649 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑦)
3530adantr 481 . . . . . . . . . . . . 13 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑦𝑋)
3634, 35sstrd 3899 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ⊆ 𝑋)
37 snex 5223 . . . . . . . . . . . . 13 {𝑥} ∈ V
3837elpw 4459 . . . . . . . . . . . 12 ({𝑥} ∈ 𝒫 𝑋 ↔ {𝑥} ⊆ 𝑋)
3936, 38sylibr 235 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑋)
4037elpw 4459 . . . . . . . . . . . 12 ({𝑥} ∈ 𝒫 𝑦 ↔ {𝑥} ⊆ 𝑦)
4134, 40sylibr 235 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ 𝒫 𝑦)
4239, 41elind 4092 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → {𝑥} ∈ (𝒫 𝑋 ∩ 𝒫 𝑦))
43 snidg 4504 . . . . . . . . . . 11 (𝑥𝑦𝑥 ∈ {𝑥})
4443ad2antrl 724 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑥 ∈ {𝑥})
45 simpll 763 . . . . . . . . . . . 12 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝑋𝑉)
4645, 36, 18syl2anc 584 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) = 𝒫 {𝑥})
47 simprr 769 . . . . . . . . . . 11 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → 𝒫 {𝑥} ∈ 𝐴)
4846, 47eqeltrd 2883 . . . . . . . . . 10 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → (𝒫 𝑋t {𝑥}) ∈ 𝐴)
49 eleq2 2871 . . . . . . . . . . . 12 (𝑢 = {𝑥} → (𝑥𝑢𝑥 ∈ {𝑥}))
50 oveq2 7024 . . . . . . . . . . . . 13 (𝑢 = {𝑥} → (𝒫 𝑋t 𝑢) = (𝒫 𝑋t {𝑥}))
5150eleq1d 2867 . . . . . . . . . . . 12 (𝑢 = {𝑥} → ((𝒫 𝑋t 𝑢) ∈ 𝐴 ↔ (𝒫 𝑋t {𝑥}) ∈ 𝐴))
5249, 51anbi12d 630 . . . . . . . . . . 11 (𝑢 = {𝑥} → ((𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴) ↔ (𝑥 ∈ {𝑥} ∧ (𝒫 𝑋t {𝑥}) ∈ 𝐴)))
5352rspcev 3559 . . . . . . . . . 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 833 . . . . . . . . 9 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ (𝑥𝑦 ∧ 𝒫 {𝑥} ∈ 𝐴)) → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5554expr 457 . . . . . . . 8 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ 𝑥𝑦) → (𝒫 {𝑥} ∈ 𝐴 → ∃𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
5655ralimdva 3144 . . . . . . 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 407 . . . . 5 (((𝑋𝑉𝑦 ∈ 𝒫 𝑋) ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
5958an32s 648 . . . 4 (((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) ∧ 𝑦 ∈ 𝒫 𝑋) → ∀𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
6059ralrimiva 3149 . . 3 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴))
61 islly 21760 . . 3 (𝒫 𝑋 ∈ Locally 𝐴 ↔ (𝒫 𝑋 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑋𝑥𝑦𝑢 ∈ (𝒫 𝑋 ∩ 𝒫 𝑦)(𝑥𝑢 ∧ (𝒫 𝑋t 𝑢) ∈ 𝐴)))
6228, 60, 61sylanbrc 583 . 2 ((𝑋𝑉 ∧ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴) → 𝒫 𝑋 ∈ Locally 𝐴)
6326, 62impbida 797 1 (𝑋𝑉 → (𝒫 𝑋 ∈ Locally 𝐴 ↔ ∀𝑥𝑋 𝒫 {𝑥} ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wrex 3106  cin 3858  wss 3859  𝒫 cpw 4453  {csn 4472  (class class class)co 7016  t crest 16523  Topctop 21185  Locally clly 21756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-oadd 7957  df-er 8139  df-en 8358  df-fin 8361  df-fi 8721  df-rest 16525  df-topgen 16546  df-top 21186  df-topon 21203  df-bases 21238  df-lly 21758
This theorem is referenced by:  disllycmp  21790  dis1stc  21791
  Copyright terms: Public domain W3C validator