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Theorem pptbas 21708
 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 21707 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴))
2 topontop 21613 . . . 4 ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top)
31, 2syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ Top)
4 eleq2 2840 . . . . . . 7 (𝑦 = {𝑥, 𝑃} → (𝑃𝑦𝑃 ∈ {𝑥, 𝑃}))
5 eqeq1 2762 . . . . . . 7 (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
64, 5orbi12d 916 . . . . . 6 (𝑦 = {𝑥, 𝑃} → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
7 simpr 488 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
8 simplr 768 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃𝐴)
97, 8prssd 4712 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
10 prex 5301 . . . . . . . 8 {𝑥, 𝑃} ∈ V
1110elpw 4498 . . . . . . 7 ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
129, 11sylibr 237 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
13 prid2g 4654 . . . . . . . 8 (𝑃𝐴𝑃 ∈ {𝑥, 𝑃})
1413ad2antlr 726 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃 ∈ {𝑥, 𝑃})
1514orcd 870 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
166, 12, 15elrabd 3604 . . . . 5 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
1716fmpttd 6870 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑥𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
1817frnd 6505 . . 3 ((𝐴𝑉𝑃𝐴) → ran (𝑥𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
19 eleq2 2840 . . . . . . 7 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
20 eqeq1 2762 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
2119, 20orbi12d 916 . . . . . 6 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑧𝑧 = ∅)))
2221elrab 3602 . . . . 5 (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))
23 elpwi 4503 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
2423ad2antrl 727 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → 𝑧𝐴)
2524sselda 3892 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝐴)
26 prid1g 4653 . . . . . . . . . 10 (𝑤𝑧𝑤 ∈ {𝑤, 𝑃})
2726adantl 485 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤 ∈ {𝑤, 𝑃})
28 simpr 488 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝑧)
29 n0i 4232 . . . . . . . . . . . 12 (𝑤𝑧 → ¬ 𝑧 = ∅)
3029adantl 485 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ¬ 𝑧 = ∅)
31 simplrr 777 . . . . . . . . . . . 12 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (𝑃𝑧𝑧 = ∅))
3231ord 861 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (¬ 𝑃𝑧𝑧 = ∅))
3330, 32mt3d 150 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑃𝑧)
3428, 33prssd 4712 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
35 preq1 4626 . . . . . . . . . . . 12 (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
3635eleq2d 2837 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
3735sseq1d 3923 . . . . . . . . . . 11 (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
3836, 37anbi12d 633 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
3938rspcev 3541 . . . . . . . . 9 ((𝑤𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
4025, 27, 34, 39syl12anc 835 . . . . . . . 8 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
4110rgenw 3082 . . . . . . . . 9 𝑥𝐴 {𝑥, 𝑃} ∈ V
42 eqid 2758 . . . . . . . . . 10 (𝑥𝐴 ↦ {𝑥, 𝑃}) = (𝑥𝐴 ↦ {𝑥, 𝑃})
43 eleq2 2840 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑤𝑣𝑤 ∈ {𝑥, 𝑃}))
44 sseq1 3917 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑣𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
4543, 44anbi12d 633 . . . . . . . . . 10 (𝑣 = {𝑥, 𝑃} → ((𝑤𝑣𝑣𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
4642, 45rexrnmptw 6852 . . . . . . . . 9 (∀𝑥𝐴 {𝑥, 𝑃} ∈ V → (∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧) ↔ ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
4741, 46ax-mp 5 . . . . . . . 8 (∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧) ↔ ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
4840, 47sylibr 237 . . . . . . 7 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
4948ralrimiva 3113 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
5049ex 416 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)) → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5122, 50syl5bi 245 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5251ralrimiv 3112 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)}∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
53 basgen2 21689 . . 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 = ∅)})
543, 18, 52, 53syl3anc 1368 . 2 ((𝐴𝑉𝑃𝐴) → (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
55 eleq2 2840 . . . 4 (𝑦 = 𝑥 → (𝑃𝑦𝑃𝑥))
56 eqeq1 2762 . . . 4 (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
5755, 56orbi12d 916 . . 3 (𝑦 = 𝑥 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑥𝑥 = ∅)))
5857cbvrabv 3404 . 2 {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}
5954, 58eqtr2di 2810 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)} = (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  𝒫 cpw 4494  {cpr 4524   ↦ cmpt 5112  ran crn 5525  ‘cfv 6335  topGenctg 16769  Topctop 21593  TopOnctopon 21610 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-topgen 16775  df-top 21594  df-topon 21611 This theorem is referenced by: (None)
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