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Theorem epttop 22375
Description: The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
epttop ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ (TopOnβ€˜π΄))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑃
Allowed substitution hint:   𝑉(π‘₯)

Proof of Theorem epttop
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab 4035 . . . . 5 (𝑦 βŠ† {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ↔ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)))
2 eleq2 2827 . . . . . . . 8 (π‘₯ = βˆͺ 𝑦 β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ βˆͺ 𝑦))
3 eqeq1 2741 . . . . . . . 8 (π‘₯ = βˆͺ 𝑦 β†’ (π‘₯ = 𝐴 ↔ βˆͺ 𝑦 = 𝐴))
42, 3imbi12d 345 . . . . . . 7 (π‘₯ = βˆͺ 𝑦 β†’ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ↔ (𝑃 ∈ βˆͺ 𝑦 β†’ βˆͺ 𝑦 = 𝐴)))
5 simprl 770 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ 𝑦 βŠ† 𝒫 𝐴)
6 sspwuni 5065 . . . . . . . . 9 (𝑦 βŠ† 𝒫 𝐴 ↔ βˆͺ 𝑦 βŠ† 𝐴)
75, 6sylib 217 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ βˆͺ 𝑦 βŠ† 𝐴)
8 vuniex 7681 . . . . . . . . 9 βˆͺ 𝑦 ∈ V
98elpw 4569 . . . . . . . 8 (βˆͺ 𝑦 ∈ 𝒫 𝐴 ↔ βˆͺ 𝑦 βŠ† 𝐴)
107, 9sylibr 233 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ βˆͺ 𝑦 ∈ 𝒫 𝐴)
11 eluni2 4874 . . . . . . . . . 10 (𝑃 ∈ βˆͺ 𝑦 ↔ βˆƒπ‘₯ ∈ 𝑦 𝑃 ∈ π‘₯)
12 r19.29 3118 . . . . . . . . . . . . 13 ((βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ βˆƒπ‘₯ ∈ 𝑦 𝑃 ∈ π‘₯) β†’ βˆƒπ‘₯ ∈ 𝑦 ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ 𝑃 ∈ π‘₯))
13 simpr 486 . . . . . . . . . . . . . . . 16 ((π‘₯ ∈ 𝑦 ∧ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)) β†’ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))
1413impr 456 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ 𝑦 ∧ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ 𝑃 ∈ π‘₯)) β†’ π‘₯ = 𝐴)
15 elssuni 4903 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ 𝑦 β†’ π‘₯ βŠ† βˆͺ 𝑦)
1615adantr 482 . . . . . . . . . . . . . . 15 ((π‘₯ ∈ 𝑦 ∧ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ 𝑃 ∈ π‘₯)) β†’ π‘₯ βŠ† βˆͺ 𝑦)
1714, 16eqsstrrd 3988 . . . . . . . . . . . . . 14 ((π‘₯ ∈ 𝑦 ∧ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ 𝑃 ∈ π‘₯)) β†’ 𝐴 βŠ† βˆͺ 𝑦)
1817rexlimiva 3145 . . . . . . . . . . . . 13 (βˆƒπ‘₯ ∈ 𝑦 ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ 𝑃 ∈ π‘₯) β†’ 𝐴 βŠ† βˆͺ 𝑦)
1912, 18syl 17 . . . . . . . . . . . 12 ((βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ∧ βˆƒπ‘₯ ∈ 𝑦 𝑃 ∈ π‘₯) β†’ 𝐴 βŠ† βˆͺ 𝑦)
2019ex 414 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) β†’ (βˆƒπ‘₯ ∈ 𝑦 𝑃 ∈ π‘₯ β†’ 𝐴 βŠ† βˆͺ 𝑦))
2120ad2antll 728 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ (βˆƒπ‘₯ ∈ 𝑦 𝑃 ∈ π‘₯ β†’ 𝐴 βŠ† βˆͺ 𝑦))
2211, 21biimtrid 241 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ (𝑃 ∈ βˆͺ 𝑦 β†’ 𝐴 βŠ† βˆͺ 𝑦))
2322, 7jctild 527 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ (𝑃 ∈ βˆͺ 𝑦 β†’ (βˆͺ 𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦)))
24 eqss 3964 . . . . . . . 8 (βˆͺ 𝑦 = 𝐴 ↔ (βˆͺ 𝑦 βŠ† 𝐴 ∧ 𝐴 βŠ† βˆͺ 𝑦))
2523, 24syl6ibr 252 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ (𝑃 ∈ βˆͺ 𝑦 β†’ βˆͺ 𝑦 = 𝐴))
264, 10, 25elrabd 3652 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴))) β†’ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})
2726ex 414 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑦 βŠ† 𝒫 𝐴 ∧ βˆ€π‘₯ ∈ 𝑦 (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)) β†’ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}))
281, 27biimtrid 241 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (𝑦 βŠ† {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} β†’ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}))
2928alrimiv 1931 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ βˆ€π‘¦(𝑦 βŠ† {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} β†’ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}))
30 inss1 4193 . . . . . . . . 9 (𝑦 ∩ 𝑧) βŠ† 𝑦
31 simprll 778 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ 𝑦 ∈ 𝒫 𝐴)
3231elpwid 4574 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ 𝑦 βŠ† 𝐴)
3330, 32sstrid 3960 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ (𝑦 ∩ 𝑧) βŠ† 𝐴)
34 vex 3452 . . . . . . . . . 10 𝑦 ∈ V
3534inex1 5279 . . . . . . . . 9 (𝑦 ∩ 𝑧) ∈ V
3635elpw 4569 . . . . . . . 8 ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) βŠ† 𝐴)
3733, 36sylibr 233 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
38 elin 3931 . . . . . . . 8 (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
39 simprlr 779 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴))
40 simprrr 781 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴))
4139, 40anim12d 610 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ ((𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) β†’ (𝑦 = 𝐴 ∧ 𝑧 = 𝐴)))
42 ineq12 4172 . . . . . . . . . 10 ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) β†’ (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝐴))
43 inidm 4183 . . . . . . . . . 10 (𝐴 ∩ 𝐴) = 𝐴
4442, 43eqtrdi 2793 . . . . . . . . 9 ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) β†’ (𝑦 ∩ 𝑧) = 𝐴)
4541, 44syl6 35 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ ((𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) β†’ (𝑦 ∩ 𝑧) = 𝐴))
4638, 45biimtrid 241 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ (𝑃 ∈ (𝑦 ∩ 𝑧) β†’ (𝑦 ∩ 𝑧) = 𝐴))
4737, 46jca 513 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))) β†’ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) β†’ (𝑦 ∩ 𝑧) = 𝐴)))
4847ex 414 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴))) β†’ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) β†’ (𝑦 ∩ 𝑧) = 𝐴))))
49 eleq2 2827 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ 𝑦))
50 eqeq1 2741 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π‘₯ = 𝐴 ↔ 𝑦 = 𝐴))
5149, 50imbi12d 345 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ↔ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)))
5251elrab 3650 . . . . . 6 (𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)))
53 eleq2 2827 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ 𝑧))
54 eqeq1 2741 . . . . . . . 8 (π‘₯ = 𝑧 β†’ (π‘₯ = 𝐴 ↔ 𝑧 = 𝐴))
5553, 54imbi12d 345 . . . . . . 7 (π‘₯ = 𝑧 β†’ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ↔ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))
5655elrab 3650 . . . . . 6 (𝑧 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴)))
5752, 56anbi12i 628 . . . . 5 ((𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}) ↔ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 β†’ 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 β†’ 𝑧 = 𝐴))))
58 eleq2 2827 . . . . . . 7 (π‘₯ = (𝑦 ∩ 𝑧) β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
59 eqeq1 2741 . . . . . . 7 (π‘₯ = (𝑦 ∩ 𝑧) β†’ (π‘₯ = 𝐴 ↔ (𝑦 ∩ 𝑧) = 𝐴))
6058, 59imbi12d 345 . . . . . 6 (π‘₯ = (𝑦 ∩ 𝑧) β†’ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) β†’ (𝑦 ∩ 𝑧) = 𝐴)))
6160elrab 3650 . . . . 5 ((𝑦 ∩ 𝑧) ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ↔ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) β†’ (𝑦 ∩ 𝑧) = 𝐴)))
6248, 57, 613imtr4g 296 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ ((𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∧ 𝑧 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}) β†’ (𝑦 ∩ 𝑧) ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}))
6362ralrimivv 3196 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ βˆ€π‘¦ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}βˆ€π‘§ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} (𝑦 ∩ 𝑧) ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})
64 pwexg 5338 . . . . . 6 (𝐴 ∈ 𝑉 β†’ 𝒫 𝐴 ∈ V)
6564adantr 482 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ 𝒫 𝐴 ∈ V)
66 rabexg 5293 . . . . 5 (𝒫 𝐴 ∈ V β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ V)
6765, 66syl 17 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ V)
68 istopg 22260 . . . 4 ({π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ V β†’ ({π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ Top ↔ (βˆ€π‘¦(𝑦 βŠ† {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} β†’ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}) ∧ βˆ€π‘¦ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}βˆ€π‘§ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} (𝑦 ∩ 𝑧) ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})))
6967, 68syl 17 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ ({π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ Top ↔ (βˆ€π‘¦(𝑦 βŠ† {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} β†’ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}) ∧ βˆ€π‘¦ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}βˆ€π‘§ ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} (𝑦 ∩ 𝑧) ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})))
7029, 63, 69mpbir2and 712 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ Top)
71 eleq2 2827 . . . . . 6 (π‘₯ = 𝐴 β†’ (𝑃 ∈ π‘₯ ↔ 𝑃 ∈ 𝐴))
72 eqeq1 2741 . . . . . 6 (π‘₯ = 𝐴 β†’ (π‘₯ = 𝐴 ↔ 𝐴 = 𝐴))
7371, 72imbi12d 345 . . . . 5 (π‘₯ = 𝐴 β†’ ((𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴) ↔ (𝑃 ∈ 𝐴 β†’ 𝐴 = 𝐴)))
74 pwidg 4585 . . . . . 6 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ 𝒫 𝐴)
7574adantr 482 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ 𝐴 ∈ 𝒫 𝐴)
76 eqidd 2738 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ 𝐴 = 𝐴)
7776a1d 25 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ (𝑃 ∈ 𝐴 β†’ 𝐴 = 𝐴))
7873, 75, 77elrabd 3652 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ 𝐴 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})
79 elssuni 4903 . . . 4 (𝐴 ∈ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} β†’ 𝐴 βŠ† βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})
8078, 79syl 17 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ 𝐴 βŠ† βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})
81 ssrab2 4042 . . . . 5 {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} βŠ† 𝒫 𝐴
82 sspwuni 5065 . . . . 5 ({π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} βŠ† 𝒫 𝐴 ↔ βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} βŠ† 𝐴)
8381, 82mpbi 229 . . . 4 βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} βŠ† 𝐴
8483a1i 11 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} βŠ† 𝐴)
8580, 84eqssd 3966 . 2 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ 𝐴 = βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)})
86 istopon 22277 . 2 ({π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ (TopOnβ€˜π΄) ↔ ({π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ Top ∧ 𝐴 = βˆͺ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)}))
8770, 85, 86sylanbrc 584 1 ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) β†’ {π‘₯ ∈ 𝒫 𝐴 ∣ (𝑃 ∈ π‘₯ β†’ π‘₯ = 𝐴)} ∈ (TopOnβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  β€˜cfv 6501  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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-top 22259  df-topon 22276
This theorem is referenced by:  dfac14lem  22984  dfac14  22985
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