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Theorem pptbas 22374
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 22373 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ∈ (TopOnβ€˜π΄))
2 topontop 22278 . . . 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 2823 . . . . . . 7 (𝑦 = {π‘₯, 𝑃} β†’ (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ {π‘₯, 𝑃}))
5 eqeq1 2737 . . . . . . 7 (𝑦 = {π‘₯, 𝑃} β†’ (𝑦 = βˆ… ↔ {π‘₯, 𝑃} = βˆ…))
64, 5orbi12d 918 . . . . . 6 (𝑦 = {π‘₯, 𝑃} β†’ ((𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…) ↔ (𝑃 ∈ {π‘₯, 𝑃} ∨ {π‘₯, 𝑃} = βˆ…)))
7 simpr 486 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ π‘₯ ∈ 𝐴)
8 simplr 768 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ 𝑃 ∈ 𝐴)
97, 8prssd 4783 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯, 𝑃} βŠ† 𝐴)
10 prex 5390 . . . . . . . 8 {π‘₯, 𝑃} ∈ V
1110elpw 4565 . . . . . . 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 4723 . . . . . . . 8 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ {π‘₯, 𝑃})
1413ad2antlr 726 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ 𝑃 ∈ {π‘₯, 𝑃})
1514orcd 872 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ (𝑃 ∈ {π‘₯, 𝑃} ∨ {π‘₯, 𝑃} = βˆ…))
166, 12, 15elrabd 3648 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ π‘₯ ∈ 𝐴) β†’ {π‘₯, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
1716fmpttd 7064 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}):𝐴⟢{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
1817frnd 6677 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}) βŠ† {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
19 eleq2 2823 . . . . . . 7 (𝑦 = 𝑧 β†’ (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
20 eqeq1 2737 . . . . . . 7 (𝑦 = 𝑧 β†’ (𝑦 = βˆ… ↔ 𝑧 = βˆ…))
2119, 20orbi12d 918 . . . . . 6 (𝑦 = 𝑧 β†’ ((𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…)))
2221elrab 3646 . . . . 5 (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…)))
23 elpwi 4568 . . . . . . . . . . 11 (𝑧 ∈ 𝒫 𝐴 β†’ 𝑧 βŠ† 𝐴)
2423ad2antrl 727 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) β†’ 𝑧 βŠ† 𝐴)
2524sselda 3945 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑀 ∈ 𝐴)
26 prid1g 4722 . . . . . . . . . 10 (𝑀 ∈ 𝑧 β†’ 𝑀 ∈ {𝑀, 𝑃})
2726adantl 483 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑀 ∈ {𝑀, 𝑃})
28 simpr 486 . . . . . . . . . 10 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ 𝑀 ∈ 𝑧)
29 n0i 4294 . . . . . . . . . . . 12 (𝑀 ∈ 𝑧 β†’ Β¬ 𝑧 = βˆ…)
3029adantl 483 . . . . . . . . . . 11 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ Β¬ 𝑧 = βˆ…)
31 simplrr 777 . . . . . . . . . . . 12 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))
3231ord 863 . . . . . . . . . . 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 4783 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = βˆ…))) ∧ 𝑀 ∈ 𝑧) β†’ {𝑀, 𝑃} βŠ† 𝑧)
35 preq1 4695 . . . . . . . . . . . 12 (π‘₯ = 𝑀 β†’ {π‘₯, 𝑃} = {𝑀, 𝑃})
3635eleq2d 2820 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ (𝑀 ∈ {π‘₯, 𝑃} ↔ 𝑀 ∈ {𝑀, 𝑃}))
3735sseq1d 3976 . . . . . . . . . . 11 (π‘₯ = 𝑀 β†’ ({π‘₯, 𝑃} βŠ† 𝑧 ↔ {𝑀, 𝑃} βŠ† 𝑧))
3836, 37anbi12d 632 . . . . . . . . . 10 (π‘₯ = 𝑀 β†’ ((𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧) ↔ (𝑀 ∈ {𝑀, 𝑃} ∧ {𝑀, 𝑃} βŠ† 𝑧)))
3938rspcev 3580 . . . . . . . . 9 ((𝑀 ∈ 𝐴 ∧ (𝑀 ∈ {𝑀, 𝑃} ∧ {𝑀, 𝑃} βŠ† 𝑧)) β†’ βˆƒπ‘₯ ∈ 𝐴 (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧))
4025, 27, 34, 39syl12anc 836 . . . . . . . 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 2733 . . . . . . . . . 10 (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃}) = (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})
43 eleq2 2823 . . . . . . . . . . 11 (𝑣 = {π‘₯, 𝑃} β†’ (𝑀 ∈ 𝑣 ↔ 𝑀 ∈ {π‘₯, 𝑃}))
44 sseq1 3970 . . . . . . . . . . 11 (𝑣 = {π‘₯, 𝑃} β†’ (𝑣 βŠ† 𝑧 ↔ {π‘₯, 𝑃} βŠ† 𝑧))
4543, 44anbi12d 632 . . . . . . . . . 10 (𝑣 = {π‘₯, 𝑃} β†’ ((𝑀 ∈ 𝑣 ∧ 𝑣 βŠ† 𝑧) ↔ (𝑀 ∈ {π‘₯, 𝑃} ∧ {π‘₯, 𝑃} βŠ† 𝑧)))
4642, 45rexrnmptw 7046 . . . . . . . . 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 3140 . . . . . 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 414 . . . . 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 3139 . . 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 22355 . . 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 1372 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝐴 ↦ {π‘₯, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)})
55 eleq2 2823 . . . 4 (𝑦 = π‘₯ β†’ (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ π‘₯))
56 eqeq1 2737 . . . 4 (𝑦 = π‘₯ β†’ (𝑦 = βˆ… ↔ π‘₯ = βˆ…))
5755, 56orbi12d 918 . . 3 (𝑦 = π‘₯ β†’ ((𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…) ↔ (𝑃 ∈ π‘₯ ∨ π‘₯ = βˆ…)))
5857cbvrabv 3416 . 2 {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = βˆ…)} = {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ ∨ π‘₯ = βˆ…)}
5954, 58eqtr2di 2790 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 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3444   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  {cpr 4589   ↦ cmpt 5189  ran crn 5635  β€˜cfv 6497  topGenctg 17324  Topctop 22258  TopOnctopon 22275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-topgen 17330  df-top 22259  df-topon 22276
This theorem is referenced by: (None)
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