MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  epttop Structured version   Visualization version   GIF version

Theorem epttop 20723
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 3659 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)))
2 simprl 793 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
3 sspwuni 4577 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝐴 𝑦𝐴)
42, 3sylib 208 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦𝐴)
5 vuniex 6907 . . . . . . . . 9 𝑦 ∈ V
65elpw 4136 . . . . . . . 8 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
74, 6sylibr 224 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ 𝒫 𝐴)
8 eluni2 4406 . . . . . . . . . 10 (𝑃 𝑦 ↔ ∃𝑥𝑦 𝑃𝑥)
9 r19.29 3065 . . . . . . . . . . . . 13 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → ∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥))
10 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ (𝑃𝑥𝑥 = 𝐴)) → (𝑃𝑥𝑥 = 𝐴))
1110impr 648 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 = 𝐴)
12 elssuni 4433 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑥 𝑦)
1312adantr 481 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 𝑦)
1411, 13eqsstr3d 3619 . . . . . . . . . . . . . 14 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝐴 𝑦)
1514rexlimiva 3021 . . . . . . . . . . . . 13 (∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥) → 𝐴 𝑦)
169, 15syl 17 . . . . . . . . . . . 12 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → 𝐴 𝑦)
1716ex 450 . . . . . . . . . . 11 (∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
1817ad2antll 764 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
198, 18syl5bi 232 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦𝐴 𝑦))
2019, 4jctild 565 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 → ( 𝑦𝐴𝐴 𝑦)))
21 eqss 3598 . . . . . . . 8 ( 𝑦 = 𝐴 ↔ ( 𝑦𝐴𝐴 𝑦))
2220, 21syl6ibr 242 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 𝑦 = 𝐴))
23 eleq2 2687 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃𝑥𝑃 𝑦))
24 eqeq1 2625 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = 𝐴 𝑦 = 𝐴))
2523, 24imbi12d 334 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 𝑦 𝑦 = 𝐴)))
2625elrab 3346 . . . . . . 7 ( 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ( 𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 𝑦 𝑦 = 𝐴)))
277, 22, 26sylanbrc 697 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
2827ex 450 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
291, 28syl5bi 232 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
3029alrimiv 1852 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
31 inss1 3811 . . . . . . . . 9 (𝑦𝑧) ⊆ 𝑦
32 simprll 801 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦 ∈ 𝒫 𝐴)
3332elpwid 4141 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦𝐴)
3431, 33syl5ss 3594 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ⊆ 𝐴)
35 vex 3189 . . . . . . . . . 10 𝑦 ∈ V
3635inex1 4759 . . . . . . . . 9 (𝑦𝑧) ∈ V
3736elpw 4136 . . . . . . . 8 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
3834, 37sylibr 224 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ∈ 𝒫 𝐴)
39 elin 3774 . . . . . . . 8 (𝑃 ∈ (𝑦𝑧) ↔ (𝑃𝑦𝑃𝑧))
40 simprlr 802 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑦𝑦 = 𝐴))
41 simprrr 804 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑧𝑧 = 𝐴))
4240, 41anim12d 585 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦 = 𝐴𝑧 = 𝐴)))
43 ineq12 3787 . . . . . . . . . 10 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = (𝐴𝐴))
44 inidm 3800 . . . . . . . . . 10 (𝐴𝐴) = 𝐴
4543, 44syl6eq 2671 . . . . . . . . 9 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = 𝐴)
4642, 45syl6 35 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦𝑧) = 𝐴))
4739, 46syl5bi 232 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))
4838, 47jca 554 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
4948ex 450 . . . . 5 ((𝐴𝑉𝑃𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))))
50 eleq2 2687 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
51 eqeq1 2625 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
5250, 51imbi12d 334 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑦𝑦 = 𝐴)))
5352elrab 3346 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)))
54 eleq2 2687 . . . . . . . 8 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
55 eqeq1 2625 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
5654, 55imbi12d 334 . . . . . . 7 (𝑥 = 𝑧 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑧𝑧 = 𝐴)))
5756elrab 3346 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))
5853, 57anbi12i 732 . . . . 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 2687 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑃𝑥𝑃 ∈ (𝑦𝑧)))
60 eqeq1 2625 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑥 = 𝐴 ↔ (𝑦𝑧) = 𝐴))
6159, 60imbi12d 334 . . . . . 6 (𝑥 = (𝑦𝑧) → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6261elrab 3346 . . . . 5 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6349, 58, 623imtr4g 285 . . . 4 ((𝐴𝑉𝑃𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
6463ralrimivv 2964 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
65 pwexg 4810 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
6665adantr 481 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝒫 𝐴 ∈ V)
67 rabexg 4772 . . . . 5 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
6866, 67syl 17 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
69 istopg 20625 . . . 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 956 . 2 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top)
72 pwidg 4144 . . . . . 6 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
7372adantr 481 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ 𝒫 𝐴)
74 eqidd 2622 . . . . . 6 ((𝐴𝑉𝑃𝐴) → 𝐴 = 𝐴)
7574a1d 25 . . . . 5 ((𝐴𝑉𝑃𝐴) → (𝑃𝐴𝐴 = 𝐴))
76 eleq2 2687 . . . . . . 7 (𝑥 = 𝐴 → (𝑃𝑥𝑃𝐴))
77 eqeq1 2625 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
7876, 77imbi12d 334 . . . . . 6 (𝑥 = 𝐴 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝐴𝐴 = 𝐴)))
7978elrab 3346 . . . . 5 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (𝑃𝐴𝐴 = 𝐴)))
8073, 75, 79sylanbrc 697 . . . 4 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
81 elssuni 4433 . . . 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 3666 . . . . 5 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴
84 sspwuni 4577 . . . . 5 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8583, 84mpbi 220 . . . 4 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴
8685a1i 11 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8782, 86eqssd 3600 . 2 ((𝐴𝑉𝑃𝐴) → 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
88 istopon 20640 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
8971, 87, 88sylanbrc 697 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  cfv 5847  Topctop 20617  TopOnctopon 20618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-top 20621  df-topon 20623
This theorem is referenced by:  dfac14lem  21330  dfac14  21331
  Copyright terms: Public domain W3C validator