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Theorem epttop 20623
 Description: The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
epttop ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem epttop
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab 3643 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)))
2 simprl 790 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
3 sspwuni 4547 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝐴 𝑦𝐴)
42, 3sylib 207 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦𝐴)
5 vuniex 6852 . . . . . . . . 9 𝑦 ∈ V
65elpw 4114 . . . . . . . 8 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
74, 6sylibr 223 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ 𝒫 𝐴)
8 eluni2 4376 . . . . . . . . . 10 (𝑃 𝑦 ↔ ∃𝑥𝑦 𝑃𝑥)
9 r19.29 3054 . . . . . . . . . . . . 13 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → ∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥))
10 simpr 476 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ (𝑃𝑥𝑥 = 𝐴)) → (𝑃𝑥𝑥 = 𝐴))
1110impr 647 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 = 𝐴)
12 elssuni 4403 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑥 𝑦)
1312adantr 480 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 𝑦)
1411, 13eqsstr3d 3603 . . . . . . . . . . . . . 14 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝐴 𝑦)
1514rexlimiva 3010 . . . . . . . . . . . . 13 (∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥) → 𝐴 𝑦)
169, 15syl 17 . . . . . . . . . . . 12 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → 𝐴 𝑦)
1716ex 449 . . . . . . . . . . 11 (∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
1817ad2antll 761 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
198, 18syl5bi 231 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦𝐴 𝑦))
2019, 4jctild 564 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 → ( 𝑦𝐴𝐴 𝑦)))
21 eqss 3583 . . . . . . . 8 ( 𝑦 = 𝐴 ↔ ( 𝑦𝐴𝐴 𝑦))
2220, 21syl6ibr 241 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 𝑦 = 𝐴))
23 eleq2 2677 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃𝑥𝑃 𝑦))
24 eqeq1 2614 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = 𝐴 𝑦 = 𝐴))
2523, 24imbi12d 333 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 𝑦 𝑦 = 𝐴)))
2625elrab 3331 . . . . . . 7 ( 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ( 𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 𝑦 𝑦 = 𝐴)))
277, 22, 26sylanbrc 695 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
2827ex 449 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
291, 28syl5bi 231 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
3029alrimiv 1842 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
31 inss1 3795 . . . . . . . . 9 (𝑦𝑧) ⊆ 𝑦
32 simprll 798 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦 ∈ 𝒫 𝐴)
3332elpwid 4118 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦𝐴)
3431, 33syl5ss 3579 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ⊆ 𝐴)
35 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
3635inex1 4727 . . . . . . . . 9 (𝑦𝑧) ∈ V
3736elpw 4114 . . . . . . . 8 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
3834, 37sylibr 223 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ∈ 𝒫 𝐴)
39 elin 3758 . . . . . . . 8 (𝑃 ∈ (𝑦𝑧) ↔ (𝑃𝑦𝑃𝑧))
40 simprlr 799 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑦𝑦 = 𝐴))
41 simprrr 801 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑧𝑧 = 𝐴))
4240, 41anim12d 584 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦 = 𝐴𝑧 = 𝐴)))
43 ineq12 3771 . . . . . . . . . 10 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = (𝐴𝐴))
44 inidm 3784 . . . . . . . . . 10 (𝐴𝐴) = 𝐴
4543, 44syl6eq 2660 . . . . . . . . 9 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = 𝐴)
4642, 45syl6 34 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦𝑧) = 𝐴))
4739, 46syl5bi 231 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))
4838, 47jca 553 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
4948ex 449 . . . . 5 ((𝐴𝑉𝑃𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))))
50 eleq2 2677 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
51 eqeq1 2614 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
5250, 51imbi12d 333 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑦𝑦 = 𝐴)))
5352elrab 3331 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)))
54 eleq2 2677 . . . . . . . 8 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
55 eqeq1 2614 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
5654, 55imbi12d 333 . . . . . . 7 (𝑥 = 𝑧 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑧𝑧 = 𝐴)))
5756elrab 3331 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))
5853, 57anbi12i 729 . . . . 5 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) ↔ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴))))
59 eleq2 2677 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑃𝑥𝑃 ∈ (𝑦𝑧)))
60 eqeq1 2614 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑥 = 𝐴 ↔ (𝑦𝑧) = 𝐴))
6159, 60imbi12d 333 . . . . . 6 (𝑥 = (𝑦𝑧) → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6261elrab 3331 . . . . 5 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6349, 58, 623imtr4g 284 . . . 4 ((𝐴𝑉𝑃𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
6463ralrimivv 2953 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
65 pwexg 4776 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
6665adantr 480 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝒫 𝐴 ∈ V)
67 rabexg 4739 . . . . 5 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
6866, 67syl 17 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
69 istopg 20525 . . . 4 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})))
7068, 69syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})))
7130, 64, 70mpbir2and 959 . 2 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top)
72 pwidg 4121 . . . . . 6 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
7372adantr 480 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ 𝒫 𝐴)
74 eqidd 2611 . . . . . 6 ((𝐴𝑉𝑃𝐴) → 𝐴 = 𝐴)
7574a1d 25 . . . . 5 ((𝐴𝑉𝑃𝐴) → (𝑃𝐴𝐴 = 𝐴))
76 eleq2 2677 . . . . . . 7 (𝑥 = 𝐴 → (𝑃𝑥𝑃𝐴))
77 eqeq1 2614 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
7876, 77imbi12d 333 . . . . . 6 (𝑥 = 𝐴 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝐴𝐴 = 𝐴)))
7978elrab 3331 . . . . 5 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (𝑃𝐴𝐴 = 𝐴)))
8073, 75, 79sylanbrc 695 . . . 4 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
81 elssuni 4403 . . . 4 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
8280, 81syl 17 . . 3 ((𝐴𝑉𝑃𝐴) → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
83 ssrab2 3650 . . . . 5 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴
84 sspwuni 4547 . . . . 5 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8583, 84mpbi 219 . . . 4 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴
8685a1i 11 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8782, 86eqssd 3585 . 2 ((𝐴𝑉𝑃𝐴) → 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
88 istopon 20540 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
8971, 87, 88sylanbrc 695 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523 This theorem is referenced by:  dfac14lem  21230  dfac14  21231
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