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Theorem pptbas 22510
Description: The particular point topology is generated by a basis consisting of pairs {π‘₯, 𝑃} for each π‘₯ ∈ 𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
pptbas ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ ∨ π‘₯ = βˆ…)} = (topGenβ€˜ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑃   π‘₯,𝑉

Proof of Theorem pptbas
Dummy variables 𝑀 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ppttop 22509 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∈ (TopOnβ€˜π΄))
2 topontop 22414 . . . 4 ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∈ (TopOnβ€˜π΄) β†’ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∈ Top)
31, 2syl 17 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∈ Top)
4 eleq2 2822 . . . . . . 7 (𝑦 = {π‘₯, 𝑃} β†’ (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ {π‘₯, 𝑃}))
5 eqeq1 2736 . . . . . . 7 (𝑦 = {π‘₯, 𝑃} β†’ (𝑦 = βˆ… ↔ {π‘₯, 𝑃} = βˆ…))
64, 5orbi12d 917 . . . . . 6 (𝑦 = {π‘₯, 𝑃} β†’ ((𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…) ↔ (𝑃 ∈ {π‘₯, 𝑃} ∨ {π‘₯, 𝑃} = βˆ…)))
7 simpr 485 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ 𝐴)
8 simplr 767 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ 𝑃 ∈ 𝐴)
97, 8prssd 4825 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯, 𝑃} βŠ† 𝐴)
10 prex 5432 . . . . . . . 8 {π‘₯, 𝑃} ∈ V
1110elpw 4606 . . . . . . 7 ({π‘₯, 𝑃} ∈ 𝒫 𝐴 ↔ {π‘₯, 𝑃} βŠ† 𝐴)
129, 11sylibr 233 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯, 𝑃} ∈ 𝒫 𝐴)
13 prid2g 4765 . . . . . . . 8 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ {π‘₯, 𝑃})
1413ad2antlr 725 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ 𝑃 ∈ {π‘₯, 𝑃})
1514orcd 871 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ (𝑃 ∈ {π‘₯, 𝑃} ∨ {π‘₯, 𝑃} = βˆ…))
166, 12, 15elrabd 3685 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
1716fmpttd 7114 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}):𝐴⟢{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
1817frnd 6725 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}) βŠ† {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
19 eleq2 2822 . . . . . . 7 (𝑦 = 𝑧 β†’ (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
20 eqeq1 2736 . . . . . . 7 (𝑦 = 𝑧 β†’ (𝑦 = βˆ… ↔ 𝑧 = βˆ…))
2119, 20orbi12d 917 . . . . . 6 (𝑦 = 𝑧 β†’ ((𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…)))
2221elrab 3683 . . . . 5 (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…)))
23 elpwi 4609 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴 β†’ 𝑧 βŠ† 𝐴)
2423ad2antrl 726 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) β†’ 𝑧 βŠ† 𝐴)
2524sselda 3982 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑀 ∈ 𝐴)
26 prid1g 4764 . . . . . . . . . 10 (𝑀 ∈ 𝑧 β†’ 𝑀 ∈ {𝑀, 𝑃})
2726adantl 482 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑀 ∈ {𝑀, 𝑃})
28 simpr 485 . . . . . . . . . 10 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑀 ∈ 𝑧)
29 n0i 4333 . . . . . . . . . . . 12 (𝑀 ∈ 𝑧 β†’ Β¬ 𝑧 = βˆ…)
3029adantl 482 . . . . . . . . . . 11 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ Β¬ 𝑧 = βˆ…)
31 simplrr 776 . . . . . . . . . . . 12 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))
3231ord 862 . . . . . . . . . . 11 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ (Β¬ 𝑃 ∈ 𝑧 β†’ 𝑧 = βˆ…))
3330, 32mt3d 148 . . . . . . . . . 10 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑃 ∈ 𝑧)
3428, 33prssd 4825 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ {𝑀, 𝑃} βŠ† 𝑧)
35 preq1 4737 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ {π‘₯, 𝑃} = {𝑀, 𝑃})
3635eleq2d 2819 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ (𝑀 ∈ {π‘₯, 𝑃} ↔ 𝑀 ∈ {𝑀, 𝑃}))
3735sseq1d 4013 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ({π‘₯, 𝑃} βŠ† 𝑧 ↔ {𝑀, 𝑃} βŠ† 𝑧))
3836, 37anbi12d 631 . . . . . . . . . 10 (π‘₯ = 𝑀 β†’ ((𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧) ↔ (𝑀 ∈ {𝑀, 𝑃} ∧ {𝑀, 𝑃} βŠ† 𝑧)))
3938rspcev 3612 . . . . . . . . 9 ((𝑀 ∈ 𝐴 ∧ (𝑀 ∈ {𝑀, 𝑃} ∧ {𝑀, 𝑃} βŠ† 𝑧)) β†’ βˆƒπ‘₯ ∈ 𝐴 (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧))
4025, 27, 34, 39syl12anc 835 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ βˆƒπ‘₯ ∈ 𝐴 (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧))
4110rgenw 3065 . . . . . . . . 9 βˆ€π‘₯ ∈ 𝐴 {π‘₯, 𝑃} ∈ V
42 eqid 2732 . . . . . . . . . 10 (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}) = (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})
43 eleq2 2822 . . . . . . . . . . 11 (𝑣 = {π‘₯, 𝑃} β†’ (𝑀 ∈ 𝑣 ↔ 𝑀 ∈ {π‘₯, 𝑃}))
44 sseq1 4007 . . . . . . . . . . 11 (𝑣 = {π‘₯, 𝑃} β†’ (𝑣 βŠ† 𝑧 ↔ {π‘₯, 𝑃} βŠ† 𝑧))
4543, 44anbi12d 631 . . . . . . . . . 10 (𝑣 = {π‘₯, 𝑃} β†’ ((𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) ↔ (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧)))
4642, 45rexrnmptw 7096 . . . . . . . . 9 (βˆ€π‘₯ ∈ 𝐴 {π‘₯, 𝑃} ∈ V β†’ (βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) ↔ βˆƒπ‘₯ ∈ 𝐴 (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧)))
4741, 46ax-mp 5 . . . . . . . 8 (βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) ↔ βˆƒπ‘₯ ∈ 𝐴 (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧))
4840, 47sylibr 233 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧))
4948ralrimiva 3146 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) β†’ βˆ€π‘€ ∈ 𝑧 βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧))
5049ex 413 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…)) β†’ βˆ€π‘€ ∈ 𝑧 βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧)))
5122, 50biimtrid 241 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} β†’ βˆ€π‘€ ∈ 𝑧 βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧)))
5251ralrimiv 3145 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ βˆ€π‘§ ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)}βˆ€π‘€ ∈ 𝑧 βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧))
53 basgen2 22491 . . 3 (({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∈ Top ∧ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}) βŠ† {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∧ βˆ€π‘§ ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)}βˆ€π‘€ ∈ 𝑧 βˆƒπ‘£ ∈ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})(𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧)) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
543, 18, 52, 53syl3anc 1371 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
55 eleq2 2822 . . . 4 (𝑦 = π‘₯ β†’ (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ π‘₯))
56 eqeq1 2736 . . . 4 (𝑦 = π‘₯ β†’ (𝑦 = βˆ… ↔ π‘₯ = βˆ…))
5755, 56orbi12d 917 . . 3 (𝑦 = π‘₯ β†’ ((𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…) ↔ (𝑃 ∈ π‘₯ ∨ π‘₯ = βˆ…)))
5857cbvrabv 3442 . 2 {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} = {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ ∨ π‘₯ = βˆ…)}
5954, 58eqtr2di 2789 1 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ ∨ π‘₯ = βˆ…)} = (topGenβ€˜ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  {cpr 4630   ↦ cmpt 5231  ran crn 5677  β€˜cfv 6543  topGenctg 17382  Topctop 22394  TopOnctopon 22411
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-topgen 17388  df-top 22395  df-topon 22412
This theorem is referenced by: (None)
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