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Theorem neibastop3 33770
Description: The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
neibastop1.1 (𝜑𝑋𝑉)
neibastop1.2 (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))
neibastop1.3 ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)
neibastop1.4 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}
neibastop1.5 ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)
neibastop1.6 ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)
Assertion
Ref Expression
neibastop3 (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
Distinct variable groups:   𝑡,𝑛,𝑣,𝑦,𝑗,𝑥   𝑗,𝐽   𝑥,𝑛,𝐽,𝑣,𝑦   𝑡,𝑜,𝑣,𝑤,𝑥,𝑦,𝑗,𝐹,𝑛   𝜑,𝑗,𝑛,𝑜,𝑡,𝑣,𝑤,𝑥,𝑦   𝑗,𝑋,𝑛,𝑜,𝑡,𝑣,𝑤,𝑥,𝑦
Allowed substitution hints:   𝐽(𝑤,𝑡,𝑜)   𝑉(𝑥,𝑦,𝑤,𝑣,𝑡,𝑗,𝑛,𝑜)

Proof of Theorem neibastop3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 neibastop1.1 . . . 4 (𝜑𝑋𝑉)
2 neibastop1.2 . . . 4 (𝜑𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))
3 neibastop1.3 . . . 4 ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥) ∧ 𝑤 ∈ (𝐹𝑥))) → ((𝐹𝑥) ∩ 𝒫 (𝑣𝑤)) ≠ ∅)
4 neibastop1.4 . . . 4 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅}
51, 2, 3, 4neibastop1 33767 . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
6 neibastop1.5 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → 𝑥𝑣)
7 neibastop1.6 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑋𝑣 ∈ (𝐹𝑥))) → ∃𝑡 ∈ (𝐹𝑥)∀𝑦𝑡 ((𝐹𝑦) ∩ 𝒫 𝑣) ≠ ∅)
81, 2, 3, 4, 6, 7neibastop2 33769 . . . . . . . 8 ((𝜑𝑧𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
9 velpw 4527 . . . . . . . . 9 (𝑛 ∈ 𝒫 𝑋𝑛𝑋)
109anbi1i 626 . . . . . . . 8 ((𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅) ↔ (𝑛𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅))
118, 10syl6bbr 292 . . . . . . 7 ((𝜑𝑧𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
1211abbi2dv 2953 . . . . . 6 ((𝜑𝑧𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)})
13 df-rab 3142 . . . . . 6 {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)}
1412, 13syl6eqr 2877 . . . . 5 ((𝜑𝑧𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
1514ralrimiva 3177 . . . 4 (𝜑 → ∀𝑧𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
16 sneq 4560 . . . . . . 7 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1716fveq2d 6665 . . . . . 6 (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
18 fveq2 6661 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1918ineq1d 4173 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹𝑥) ∩ 𝒫 𝑛) = ((𝐹𝑧) ∩ 𝒫 𝑛))
2019neeq1d 3073 . . . . . . 7 (𝑥 = 𝑧 → (((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅))
2120rabbidv 3465 . . . . . 6 (𝑥 = 𝑧 → {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
2217, 21eqeq12d 2840 . . . . 5 (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅}))
2322cbvralvw 3434 . . . 4 (∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∀𝑧𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
2415, 23sylibr 237 . . 3 (𝜑 → ∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
25 toponuni 21525 . . . . . . . . . 10 (𝑗 ∈ (TopOn‘𝑋) → 𝑋 = 𝑗)
26 eqimss2 4010 . . . . . . . . . 10 (𝑋 = 𝑗 𝑗𝑋)
2725, 26syl 17 . . . . . . . . 9 (𝑗 ∈ (TopOn‘𝑋) → 𝑗𝑋)
28 sspwuni 5008 . . . . . . . . 9 (𝑗 ⊆ 𝒫 𝑋 𝑗𝑋)
2927, 28sylibr 237 . . . . . . . 8 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ⊆ 𝒫 𝑋)
3029ad2antlr 726 . . . . . . 7 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 ⊆ 𝒫 𝑋)
31 sseqin2 4177 . . . . . . 7 (𝑗 ⊆ 𝒫 𝑋 ↔ (𝒫 𝑋𝑗) = 𝑗)
3230, 31sylib 221 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = 𝑗)
33 topontop 21524 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3433ad3antlr 730 . . . . . . . . . 10 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑗 ∈ Top)
35 eltop2 21586 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑜𝑗 ↔ ∀𝑥𝑜𝑧𝑗 (𝑥𝑧𝑧𝑜)))
3634, 35syl 17 . . . . . . . . 9 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (𝑜𝑗 ↔ ∀𝑥𝑜𝑧𝑗 (𝑥𝑧𝑧𝑜)))
37 elpwi 4531 . . . . . . . . . . . . . . 15 (𝑜 ∈ 𝒫 𝑋𝑜𝑋)
38 ssralv 4019 . . . . . . . . . . . . . . 15 (𝑜𝑋 → (∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
3937, 38syl 17 . . . . . . . . . . . . . 14 (𝑜 ∈ 𝒫 𝑋 → (∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
4039adantl 485 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
41 simprr 772 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
4241eleq2d 2901 . . . . . . . . . . . . . . . 16 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ 𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
4333ad3antlr 730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑗 ∈ Top)
4425adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → 𝑋 = 𝑗)
4544sseq2d 3985 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑜𝑋𝑜 𝑗))
4645biimpa 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜𝑋) → 𝑜 𝑗)
4737, 46sylan2 595 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑜 𝑗)
4847sselda 3953 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥𝑜) → 𝑥 𝑗)
4948adantrr 716 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑥 𝑗)
5047adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑜 𝑗)
51 eqid 2824 . . . . . . . . . . . . . . . . . . 19 𝑗 = 𝑗
5251isneip 21716 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑥 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ (𝑜 𝑗 ∧ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜))))
5352baibd 543 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑥 𝑗) ∧ 𝑜 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜)))
5443, 49, 50, 53syl21anc 836 . . . . . . . . . . . . . . . 16 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜)))
55 pweq 4538 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑜 → 𝒫 𝑛 = 𝒫 𝑜)
5655ineq2d 4174 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑜 → ((𝐹𝑥) ∩ 𝒫 𝑛) = ((𝐹𝑥) ∩ 𝒫 𝑜))
5756neeq1d 3073 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑜 → (((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
5857elrab3 3667 . . . . . . . . . . . . . . . . 17 (𝑜 ∈ 𝒫 𝑋 → (𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
5958ad2antlr 726 . . . . . . . . . . . . . . . 16 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6042, 54, 593bitr3d 312 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6160expr 460 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥𝑜) → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
6261ralimdva 3172 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
6340, 62syld 47 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
6463imp 410 . . . . . . . . . . 11 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → ∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6564an32s 651 . . . . . . . . . 10 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → ∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
66 ralbi 3162 . . . . . . . . . 10 (∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅) → (∀𝑥𝑜𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6765, 66syl 17 . . . . . . . . 9 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥𝑜𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6836, 67bitrd 282 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (𝑜𝑗 ↔ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6968rabbi2dva 4179 . . . . . . 7 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅})
7069, 4syl6eqr 2877 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = 𝐽)
7132, 70eqtr3d 2861 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽)
7271expl 461 . . . 4 (𝜑 → ((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
7372alrimiv 1929 . . 3 (𝜑 → ∀𝑗((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
74 eleq1 2903 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘𝑋)))
75 fveq2 6661 . . . . . . . 8 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
7675fveq1d 6663 . . . . . . 7 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
7776eqeq1d 2826 . . . . . 6 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
7877ralbidv 3192 . . . . 5 (𝑗 = 𝐽 → (∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
7974, 78anbi12d 633 . . . 4 (𝑗 = 𝐽 → ((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})))
8079eqeu 3683 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ ∀𝑗((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽)) → ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
815, 5, 24, 73, 80syl121anc 1372 . 2 (𝜑 → ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
82 df-reu 3140 . 2 (∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
8381, 82sylibr 237 1 (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084  wal 1536   = wceq 1538  wcel 2115  ∃!weu 2654  {cab 2802  wne 3014  wral 3133  wrex 3134  ∃!wreu 3135  {crab 3137  cdif 3916  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   cuni 4824  wf 6339  cfv 6343  Topctop 21504  TopOnctopon 21521  neicnei 21708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-topgen 16717  df-top 21505  df-topon 21522  df-nei 21709
This theorem is referenced by: (None)
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