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Theorem pptbas 20622
 Description: The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥 ∈ 𝐴. (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 20621 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴))
2 topontop 20541 . . . 4 ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top)
31, 2syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top)
4 simpr 476 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
5 simplr 788 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃𝐴)
6 prssi 4293 . . . . . . . 8 ((𝑥𝐴𝑃𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
74, 5, 6syl2anc 691 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
8 prex 4836 . . . . . . . 8 {𝑥, 𝑃} ∈ V
98elpw 4114 . . . . . . 7 ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
107, 9sylibr 223 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
11 prid2g 4240 . . . . . . . 8 (𝑃𝐴𝑃 ∈ {𝑥, 𝑃})
1211ad2antlr 759 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃 ∈ {𝑥, 𝑃})
1312orcd 406 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
14 eleq2 2677 . . . . . . . 8 (𝑦 = {𝑥, 𝑃} → (𝑃𝑦𝑃 ∈ {𝑥, 𝑃}))
15 eqeq1 2614 . . . . . . . 8 (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
1614, 15orbi12d 742 . . . . . . 7 (𝑦 = {𝑥, 𝑃} → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
1716elrab 3331 . . . . . 6 ({𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ ({𝑥, 𝑃} ∈ 𝒫 𝐴 ∧ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
1810, 13, 17sylanbrc 695 . . . . 5 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
19 eqid 2610 . . . . 5 (𝑥𝐴 ↦ {𝑥, 𝑃}) = (𝑥𝐴 ↦ {𝑥, 𝑃})
2018, 19fmptd 6292 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑥𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
21 frn 5966 . . . 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 2677 . . . . . . 7 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
24 eqeq1 2614 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
2523, 24orbi12d 742 . . . . . 6 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑧𝑧 = ∅)))
2625elrab 3331 . . . . 5 (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))
27 elpwi 4117 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
2827ad2antrl 760 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → 𝑧𝐴)
2928sselda 3568 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝐴)
30 prid1g 4239 . . . . . . . . . 10 (𝑤𝑧𝑤 ∈ {𝑤, 𝑃})
3130adantl 481 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤 ∈ {𝑤, 𝑃})
32 simpr 476 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝑧)
33 n0i 3879 . . . . . . . . . . . 12 (𝑤𝑧 → ¬ 𝑧 = ∅)
3433adantl 481 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ¬ 𝑧 = ∅)
35 simplrr 797 . . . . . . . . . . . 12 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (𝑃𝑧𝑧 = ∅))
3635ord 391 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (¬ 𝑃𝑧𝑧 = ∅))
3734, 36mt3d 139 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑃𝑧)
38 prssi 4293 . . . . . . . . . 10 ((𝑤𝑧𝑃𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
3932, 37, 38syl2anc 691 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
40 preq1 4212 . . . . . . . . . . . 12 (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
4140eleq2d 2673 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
4240sseq1d 3595 . . . . . . . . . . 11 (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
4341, 42anbi12d 743 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
4443rspcev 3282 . . . . . . . . 9 ((𝑤𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
4529, 31, 39, 44syl12anc 1316 . . . . . . . 8 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
468rgenw 2908 . . . . . . . . 9 𝑥𝐴 {𝑥, 𝑃} ∈ V
47 eleq2 2677 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑤𝑣𝑤 ∈ {𝑥, 𝑃}))
48 sseq1 3589 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑣𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
4947, 48anbi12d 743 . . . . . . . . . 10 (𝑣 = {𝑥, 𝑃} → ((𝑤𝑣𝑣𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
5019, 49rexrnmpt 6277 . . . . . . . . 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 223 . . . . . . 7 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
5352ralrimiva 2949 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
5453ex 449 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)) → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5526, 54syl5bi 231 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5655ralrimiv 2948 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)}∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
57 basgen2 20604 . . 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 1318 . 2 ((𝐴𝑉𝑃𝐴) → (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
59 eleq2 2677 . . . 4 (𝑦 = 𝑥 → (𝑃𝑦𝑃𝑥))
60 eqeq1 2614 . . . 4 (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
6159, 60orbi12d 742 . . 3 (𝑦 = 𝑥 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑥𝑥 = ∅)))
6261cbvrabv 3172 . 2 {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}
6358, 62syl6req 2661 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} = (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {cpr 4127   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  ‘cfv 5804  topGenctg 15921  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-topgen 15927  df-top 20521  df-topon 20523 This theorem is referenced by: (None)
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