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Theorem neibastop3 32663
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 32660 . . 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 32662 . . . . . . . 8 ((𝜑𝑧𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
9 selpw 4309 . . . . . . . . 9 (𝑛 ∈ 𝒫 𝑋𝑛𝑋)
109anbi1i 733 . . . . . . . 8 ((𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅) ↔ (𝑛𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅))
118, 10syl6bbr 278 . . . . . . 7 ((𝜑𝑧𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
1211abbi2dv 2880 . . . . . 6 ((𝜑𝑧𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)})
13 df-rab 3059 . . . . . 6 {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)}
1412, 13syl6eqr 2812 . . . . 5 ((𝜑𝑧𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
1514ralrimiva 3104 . . . 4 (𝜑 → ∀𝑧𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
16 sneq 4331 . . . . . . 7 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1716fveq2d 6356 . . . . . 6 (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
18 fveq2 6352 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1918ineq1d 3956 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹𝑥) ∩ 𝒫 𝑛) = ((𝐹𝑧) ∩ 𝒫 𝑛))
2019neeq1d 2991 . . . . . . 7 (𝑥 = 𝑧 → (((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅))
2120rabbidv 3329 . . . . . 6 (𝑥 = 𝑧 → {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
2217, 21eqeq12d 2775 . . . . 5 (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅}))
2322cbvralv 3310 . . . 4 (∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∀𝑧𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
2415, 23sylibr 224 . . 3 (𝜑 → ∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
25 toponuni 20921 . . . . . . . . . 10 (𝑗 ∈ (TopOn‘𝑋) → 𝑋 = 𝑗)
26 eqimss2 3799 . . . . . . . . . 10 (𝑋 = 𝑗 𝑗𝑋)
2725, 26syl 17 . . . . . . . . 9 (𝑗 ∈ (TopOn‘𝑋) → 𝑗𝑋)
28 sspwuni 4763 . . . . . . . . 9 (𝑗 ⊆ 𝒫 𝑋 𝑗𝑋)
2927, 28sylibr 224 . . . . . . . 8 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ⊆ 𝒫 𝑋)
3029ad2antlr 765 . . . . . . 7 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 ⊆ 𝒫 𝑋)
31 sseqin2 3960 . . . . . . 7 (𝑗 ⊆ 𝒫 𝑋 ↔ (𝒫 𝑋𝑗) = 𝑗)
3230, 31sylib 208 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = 𝑗)
33 topontop 20920 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3433ad3antlr 769 . . . . . . . . . 10 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑗 ∈ Top)
35 eltop2 20981 . . . . . . . . . 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 4312 . . . . . . . . . . . . . . 15 (𝑜 ∈ 𝒫 𝑋𝑜𝑋)
38 ssralv 3807 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 813 . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑗 ∈ Top)
4425adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → 𝑋 = 𝑗)
4544sseq2d 3774 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑜𝑋𝑜 𝑗))
4645biimpa 502 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜𝑋) → 𝑜 𝑗)
4737, 46sylan2 492 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑜 𝑗)
4847sselda 3744 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥𝑜) → 𝑥 𝑗)
4948adantrr 755 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑥 𝑗)
5047adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑜 𝑗)
51 eqid 2760 . . . . . . . . . . . . . . . . . . 19 𝑗 = 𝑗
5251isneip 21111 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑥 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ (𝑜 𝑗 ∧ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜))))
5352baibd 986 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑥 𝑗) ∧ 𝑜 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜)))
5443, 49, 50, 53syl21anc 1476 . . . . . . . . . . . . . . . 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 4305 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑜 → 𝒫 𝑛 = 𝒫 𝑜)
5655ineq2d 3957 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑜 → ((𝐹𝑥) ∩ 𝒫 𝑛) = ((𝐹𝑥) ∩ 𝒫 𝑜))
5756neeq1d 2991 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑜 → (((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
5857elrab3 3505 . . . . . . . . . . . . . . . . 17 (𝑜 ∈ 𝒫 𝑋 → (𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
5958ad2antlr 765 . . . . . . . . . . . . . . . 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 298 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6160expr 644 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥𝑜) → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
6261ralimdva 3100 . . . . . . . . . . . . 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 444 . . . . . . . . . . 11 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → ∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6564an32s 881 . . . . . . . . . 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 3206 . . . . . . . . . 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 268 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (𝑜𝑗 ↔ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6968rabbi2dva 3964 . . . . . . 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 2812 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = 𝐽)
7132, 70eqtr3d 2796 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽)
7271expl 649 . . . 4 (𝜑 → ((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
7372alrimiv 2004 . . 3 (𝜑 → ∀𝑗((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
74 eleq1 2827 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘𝑋)))
75 fveq2 6352 . . . . . . . 8 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
7675fveq1d 6354 . . . . . . 7 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
7776eqeq1d 2762 . . . . . 6 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
7877ralbidv 3124 . . . . 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 749 . . . 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 3518 . . 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 1482 . 2 (𝜑 → ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
82 df-reu 3057 . 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 224 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 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wcel 2139  ∃!weu 2607  {cab 2746  wne 2932  wral 3050  wrex 3051  ∃!wreu 3052  {crab 3054  cdif 3712  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588  wf 6045  cfv 6049  Topctop 20900  TopOnctopon 20917  neicnei 21103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-topgen 16306  df-top 20901  df-topon 20918  df-nei 21104
This theorem is referenced by: (None)
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