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

Theorem pptbas 20717
Description: The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
pptbas ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} = (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃   𝑥,𝑉

Proof of Theorem pptbas
Dummy variables 𝑤 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ppttop 20716 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴))
2 topontop 20636 . . . 4 ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top)
31, 2syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top)
4 simpr 477 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
5 simplr 791 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃𝐴)
6 prssi 4326 . . . . . . . 8 ((𝑥𝐴𝑃𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
74, 5, 6syl2anc 692 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
8 prex 4875 . . . . . . . 8 {𝑥, 𝑃} ∈ V
98elpw 4141 . . . . . . 7 ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
107, 9sylibr 224 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
11 prid2g 4271 . . . . . . . 8 (𝑃𝐴𝑃 ∈ {𝑥, 𝑃})
1211ad2antlr 762 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃 ∈ {𝑥, 𝑃})
1312orcd 407 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
14 eleq2 2693 . . . . . . . 8 (𝑦 = {𝑥, 𝑃} → (𝑃𝑦𝑃 ∈ {𝑥, 𝑃}))
15 eqeq1 2630 . . . . . . . 8 (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
1614, 15orbi12d 745 . . . . . . 7 (𝑦 = {𝑥, 𝑃} → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
1716elrab 3351 . . . . . 6 ({𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ ({𝑥, 𝑃} ∈ 𝒫 𝐴 ∧ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
1810, 13, 17sylanbrc 697 . . . . 5 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
19 eqid 2626 . . . . 5 (𝑥𝐴 ↦ {𝑥, 𝑃}) = (𝑥𝐴 ↦ {𝑥, 𝑃})
2018, 19fmptd 6341 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑥𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
21 frn 6012 . . . 4 ((𝑥𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} → ran (𝑥𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
2220, 21syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → ran (𝑥𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
23 eleq2 2693 . . . . . . 7 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
24 eqeq1 2630 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
2523, 24orbi12d 745 . . . . . 6 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑧𝑧 = ∅)))
2625elrab 3351 . . . . 5 (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))
27 elpwi 4145 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
2827ad2antrl 763 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → 𝑧𝐴)
2928sselda 3588 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝐴)
30 prid1g 4270 . . . . . . . . . 10 (𝑤𝑧𝑤 ∈ {𝑤, 𝑃})
3130adantl 482 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤 ∈ {𝑤, 𝑃})
32 simpr 477 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝑧)
33 n0i 3901 . . . . . . . . . . . 12 (𝑤𝑧 → ¬ 𝑧 = ∅)
3433adantl 482 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ¬ 𝑧 = ∅)
35 simplrr 800 . . . . . . . . . . . 12 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (𝑃𝑧𝑧 = ∅))
3635ord 392 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (¬ 𝑃𝑧𝑧 = ∅))
3734, 36mt3d 140 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑃𝑧)
38 prssi 4326 . . . . . . . . . 10 ((𝑤𝑧𝑃𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
3932, 37, 38syl2anc 692 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
40 preq1 4243 . . . . . . . . . . . 12 (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
4140eleq2d 2689 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
4240sseq1d 3616 . . . . . . . . . . 11 (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
4341, 42anbi12d 746 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
4443rspcev 3300 . . . . . . . . 9 ((𝑤𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
4529, 31, 39, 44syl12anc 1321 . . . . . . . 8 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
468rgenw 2924 . . . . . . . . 9 𝑥𝐴 {𝑥, 𝑃} ∈ V
47 eleq2 2693 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑤𝑣𝑤 ∈ {𝑥, 𝑃}))
48 sseq1 3610 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑣𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
4947, 48anbi12d 746 . . . . . . . . . 10 (𝑣 = {𝑥, 𝑃} → ((𝑤𝑣𝑣𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
5019, 49rexrnmpt 6326 . . . . . . . . 9 (∀𝑥𝐴 {𝑥, 𝑃} ∈ V → (∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧) ↔ ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
5146, 50ax-mp 5 . . . . . . . 8 (∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧) ↔ ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
5245, 51sylibr 224 . . . . . . 7 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
5352ralrimiva 2965 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
5453ex 450 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)) → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5526, 54syl5bi 232 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5655ralrimiv 2964 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)}∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
57 basgen2 20699 . . 3 (({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top ∧ ran (𝑥𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∧ ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)}∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)) → (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
583, 22, 56, 57syl3anc 1323 . 2 ((𝐴𝑉𝑃𝐴) → (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
59 eleq2 2693 . . . 4 (𝑦 = 𝑥 → (𝑃𝑦𝑃𝑥))
60 eqeq1 2630 . . . 4 (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
6159, 60orbi12d 745 . . 3 (𝑦 = 𝑥 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑥𝑥 = ∅)))
6261cbvrabv 3190 . 2 {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}
6358, 62syl6req 2677 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} = (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  {crab 2916  Vcvv 3191  wss 3560  c0 3896  𝒫 cpw 4135  {cpr 4155  cmpt 4678  ran crn 5080  wf 5846  cfv 5850  topGenctg 16014  Topctop 20612  TopOnctopon 20613
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  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 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-topgen 16020  df-top 20616  df-topon 20618
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator