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Theorem pptbas 20860
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 20859 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ∈ (TopOn‘𝐴))
2 topontop 20766 . . . 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 807 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃𝐴)
6 prssi 4385 . . . . . . . 8 ((𝑥𝐴𝑃𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
74, 5, 6syl2anc 694 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
8 prex 4939 . . . . . . . 8 {𝑥, 𝑃} ∈ V
98elpw 4197 . . . . . . 7 ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
107, 9sylibr 224 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
11 prid2g 4328 . . . . . . . 8 (𝑃𝐴𝑃 ∈ {𝑥, 𝑃})
1211ad2antlr 763 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → 𝑃 ∈ {𝑥, 𝑃})
1312orcd 406 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
14 eleq2 2719 . . . . . . . 8 (𝑦 = {𝑥, 𝑃} → (𝑃𝑦𝑃 ∈ {𝑥, 𝑃}))
15 eqeq1 2655 . . . . . . . 8 (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
1614, 15orbi12d 746 . . . . . . 7 (𝑦 = {𝑥, 𝑃} → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
1716elrab 3396 . . . . . 6 ({𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ ({𝑥, 𝑃} ∈ 𝒫 𝐴 ∧ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
1810, 13, 17sylanbrc 699 . . . . 5 (((𝐴𝑉𝑃𝐴) ∧ 𝑥𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
19 eqid 2651 . . . . 5 (𝑥𝐴 ↦ {𝑥, 𝑃}) = (𝑥𝐴 ↦ {𝑥, 𝑃})
2018, 19fmptd 6425 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑥𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
21 frn 6091 . . . 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 2719 . . . . . . 7 (𝑦 = 𝑧 → (𝑃𝑦𝑃𝑧))
24 eqeq1 2655 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
2523, 24orbi12d 746 . . . . . 6 (𝑦 = 𝑧 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑧𝑧 = ∅)))
2625elrab 3396 . . . . 5 (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅)))
27 elpwi 4201 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
2827ad2antrl 764 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) → 𝑧𝐴)
2928sselda 3636 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑤𝐴)
30 prid1g 4327 . . . . . . . . . 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 3953 . . . . . . . . . . . 12 (𝑤𝑧 → ¬ 𝑧 = ∅)
3433adantl 481 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ¬ 𝑧 = ∅)
35 simplrr 818 . . . . . . . . . . . 12 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (𝑃𝑧𝑧 = ∅))
3635ord 391 . . . . . . . . . . 11 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → (¬ 𝑃𝑧𝑧 = ∅))
3734, 36mt3d 140 . . . . . . . . . 10 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → 𝑃𝑧)
38 prssi 4385 . . . . . . . . . 10 ((𝑤𝑧𝑃𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
3932, 37, 38syl2anc 694 . . . . . . . . 9 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
40 preq1 4300 . . . . . . . . . . . 12 (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
4140eleq2d 2716 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
4240sseq1d 3665 . . . . . . . . . . 11 (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
4341, 42anbi12d 747 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
4443rspcev 3340 . . . . . . . . 9 ((𝑤𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
4529, 31, 39, 44syl12anc 1364 . . . . . . . 8 ((((𝐴𝑉𝑃𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = ∅))) ∧ 𝑤𝑧) → ∃𝑥𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
468rgenw 2953 . . . . . . . . 9 𝑥𝐴 {𝑥, 𝑃} ∈ V
47 eleq2 2719 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑤𝑣𝑤 ∈ {𝑥, 𝑃}))
48 sseq1 3659 . . . . . . . . . . 11 (𝑣 = {𝑥, 𝑃} → (𝑣𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
4947, 48anbi12d 747 . . . . . . . . . 10 (𝑣 = {𝑥, 𝑃} → ((𝑤𝑣𝑣𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
5019, 49rexrnmpt 6409 . . . . . . . . 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 2995 . . . . . 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 232 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} → ∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧)))
5655ralrimiv 2994 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)}∀𝑤𝑧𝑣 ∈ ran (𝑥𝐴 ↦ {𝑥, 𝑃})(𝑤𝑣𝑣𝑧))
57 basgen2 20841 . . 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 1366 . 2 ((𝐴𝑉𝑃𝐴) → (topGen‘ran (𝑥𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)})
59 eleq2 2719 . . . 4 (𝑦 = 𝑥 → (𝑃𝑦𝑃𝑥))
60 eqeq1 2655 . . . 4 (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
6159, 60orbi12d 746 . . 3 (𝑦 = 𝑥 → ((𝑃𝑦𝑦 = ∅) ↔ (𝑃𝑥𝑥 = ∅)))
6261cbvrabv 3230 . 2 {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃𝑦𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = ∅)}
6358, 62syl6req 2702 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 382  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191  {cpr 4212  cmpt 4762  ran crn 5144  wf 5922  cfv 5926  topGenctg 16145  Topctop 20746  TopOnctopon 20763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-topgen 16151  df-top 20747  df-topon 20764
This theorem is referenced by: (None)
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