ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  epttop GIF version

Theorem epttop 13257
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 3233 . . . . 5 (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)))
2 simprl 529 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
3 sspwuni 3968 . . . . . . . . 9 (𝑦 ⊆ 𝒫 𝐴 𝑦𝐴)
42, 3sylib 122 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦𝐴)
5 vuniex 4435 . . . . . . . . 9 𝑦 ∈ V
65elpw 3580 . . . . . . . 8 ( 𝑦 ∈ 𝒫 𝐴 𝑦𝐴)
74, 6sylibr 134 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ 𝒫 𝐴)
8 eluni2 3811 . . . . . . . . . 10 (𝑃 𝑦 ↔ ∃𝑥𝑦 𝑃𝑥)
9 r19.29 2614 . . . . . . . . . . . . 13 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → ∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥))
10 simpr 110 . . . . . . . . . . . . . . . 16 ((𝑥𝑦 ∧ (𝑃𝑥𝑥 = 𝐴)) → (𝑃𝑥𝑥 = 𝐴))
1110impr 379 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 = 𝐴)
12 elssuni 3835 . . . . . . . . . . . . . . . 16 (𝑥𝑦𝑥 𝑦)
1312adantr 276 . . . . . . . . . . . . . . 15 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝑥 𝑦)
1411, 13eqsstrrd 3192 . . . . . . . . . . . . . 14 ((𝑥𝑦 ∧ ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥)) → 𝐴 𝑦)
1514rexlimiva 2589 . . . . . . . . . . . . 13 (∃𝑥𝑦 ((𝑃𝑥𝑥 = 𝐴) ∧ 𝑃𝑥) → 𝐴 𝑦)
169, 15syl 14 . . . . . . . . . . . 12 ((∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) ∧ ∃𝑥𝑦 𝑃𝑥) → 𝐴 𝑦)
1716ex 115 . . . . . . . . . . 11 (∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
1817ad2antll 491 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (∃𝑥𝑦 𝑃𝑥𝐴 𝑦))
198, 18biimtrid 152 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦𝐴 𝑦))
2019, 4jctild 316 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 → ( 𝑦𝐴𝐴 𝑦)))
21 eqss 3170 . . . . . . . 8 ( 𝑦 = 𝐴 ↔ ( 𝑦𝐴𝐴 𝑦))
2220, 21syl6ibr 162 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → (𝑃 𝑦 𝑦 = 𝐴))
23 eleq2 2241 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑃𝑥𝑃 𝑦))
24 eqeq1 2184 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 = 𝐴 𝑦 = 𝐴))
2523, 24imbi12d 234 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 𝑦 𝑦 = 𝐴)))
2625elrab 2893 . . . . . . 7 ( 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ( 𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 𝑦 𝑦 = 𝐴)))
277, 22, 26sylanbrc 417 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴))) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
2827ex 115 . . . . 5 ((𝐴𝑉𝑃𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥𝑦 (𝑃𝑥𝑥 = 𝐴)) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
291, 28biimtrid 152 . . . 4 ((𝐴𝑉𝑃𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
3029alrimiv 1874 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
31 inss1 3355 . . . . . . . . 9 (𝑦𝑧) ⊆ 𝑦
32 simprll 537 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦 ∈ 𝒫 𝐴)
3332elpwid 3585 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → 𝑦𝐴)
3431, 33sstrid 3166 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ⊆ 𝐴)
35 vex 2740 . . . . . . . . . 10 𝑦 ∈ V
3635inex1 4134 . . . . . . . . 9 (𝑦𝑧) ∈ V
3736elpw 3580 . . . . . . . 8 ((𝑦𝑧) ∈ 𝒫 𝐴 ↔ (𝑦𝑧) ⊆ 𝐴)
3834, 37sylibr 134 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑦𝑧) ∈ 𝒫 𝐴)
39 elin 3318 . . . . . . . 8 (𝑃 ∈ (𝑦𝑧) ↔ (𝑃𝑦𝑃𝑧))
40 simprlr 538 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑦𝑦 = 𝐴))
41 simprrr 540 . . . . . . . . . 10 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃𝑧𝑧 = 𝐴))
4240, 41anim12d 335 . . . . . . . . 9 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦 = 𝐴𝑧 = 𝐴)))
43 ineq12 3331 . . . . . . . . . 10 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = (𝐴𝐴))
44 inidm 3344 . . . . . . . . . 10 (𝐴𝐴) = 𝐴
4543, 44eqtrdi 2226 . . . . . . . . 9 ((𝑦 = 𝐴𝑧 = 𝐴) → (𝑦𝑧) = 𝐴)
4642, 45syl6 33 . . . . . . . 8 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑃𝑦𝑃𝑧) → (𝑦𝑧) = 𝐴))
4739, 46biimtrid 152 . . . . . . 7 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))
4838, 47jca 306 . . . . . 6 (((𝐴𝑉𝑃𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
4948ex 115 . . . . 5 ((𝐴𝑉𝑃𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴))) → ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴))))
50 eleq2 2241 . . . . . . . 8 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
51 eqeq1 2184 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
5250, 51imbi12d 234 . . . . . . 7 (𝑥 = 𝑦 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑦𝑦 = 𝐴)))
5352elrab 2893 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃𝑦𝑦 = 𝐴)))
54 eleq2 2241 . . . . . . . 8 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
55 eqeq1 2184 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝐴𝑧 = 𝐴))
5654, 55imbi12d 234 . . . . . . 7 (𝑥 = 𝑧 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝑧𝑧 = 𝐴)))
5756elrab 2893 . . . . . 6 (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃𝑧𝑧 = 𝐴)))
5853, 57anbi12i 460 . . . . 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 2241 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑃𝑥𝑃 ∈ (𝑦𝑧)))
60 eqeq1 2184 . . . . . . 7 (𝑥 = (𝑦𝑧) → (𝑥 = 𝐴 ↔ (𝑦𝑧) = 𝐴))
6159, 60imbi12d 234 . . . . . 6 (𝑥 = (𝑦𝑧) → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6261elrab 2893 . . . . 5 ((𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ ((𝑦𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦𝑧) → (𝑦𝑧) = 𝐴)))
6349, 58, 623imtr4g 205 . . . 4 ((𝐴𝑉𝑃𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}) → (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
6463ralrimivv 2558 . . 3 ((𝐴𝑉𝑃𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} (𝑦𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
65 pwexg 4177 . . . . . 6 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
6665adantr 276 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝒫 𝐴 ∈ V)
67 rabexg 4143 . . . . 5 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
6866, 67syl 14 . . . 4 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ V)
69 istopg 13164 . . . 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 14 . . 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 944 . 2 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top)
72 pwidg 3588 . . . . . 6 (𝐴𝑉𝐴 ∈ 𝒫 𝐴)
7372adantr 276 . . . . 5 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ 𝒫 𝐴)
74 eqidd 2178 . . . . . 6 ((𝐴𝑉𝑃𝐴) → 𝐴 = 𝐴)
7574a1d 22 . . . . 5 ((𝐴𝑉𝑃𝐴) → (𝑃𝐴𝐴 = 𝐴))
76 eleq2 2241 . . . . . . 7 (𝑥 = 𝐴 → (𝑃𝑥𝑃𝐴))
77 eqeq1 2184 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
7876, 77imbi12d 234 . . . . . 6 (𝑥 = 𝐴 → ((𝑃𝑥𝑥 = 𝐴) ↔ (𝑃𝐴𝐴 = 𝐴)))
7978elrab 2893 . . . . 5 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (𝑃𝐴𝐴 = 𝐴)))
8073, 75, 79sylanbrc 417 . . . 4 ((𝐴𝑉𝑃𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
81 elssuni 3835 . . . 4 (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
8280, 81syl 14 . . 3 ((𝐴𝑉𝑃𝐴) → 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
83 ssrab2 3240 . . . . 5 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴
84 sspwuni 3968 . . . . 5 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝒫 𝐴 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8583, 84mpbi 145 . . . 4 {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴
8685a1i 9 . . 3 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ⊆ 𝐴)
8782, 86eqssd 3172 . 2 ((𝐴𝑉𝑃𝐴) → 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)})
88 istopon 13178 . 2 ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ Top ∧ 𝐴 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)}))
8971, 87, 88sylanbrc 417 1 ((𝐴𝑉𝑃𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃𝑥𝑥 = 𝐴)} ∈ (TopOn‘𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wcel 2148  wral 2455  wrex 2456  {crab 2459  Vcvv 2737  cin 3128  wss 3129  𝒫 cpw 3574   cuni 3807  cfv 5212  Topctop 13162  TopOnctopon 13175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-top 13163  df-topon 13176
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator