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Theorem neibastop3 32800
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 32797 . . 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 32799 . . . . . . . 8 ((𝜑𝑧𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
9 selpw 4322 . . . . . . . . 9 (𝑛 ∈ 𝒫 𝑋𝑛𝑋)
109anbi1i 617 . . . . . . . 8 ((𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅) ↔ (𝑛𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅))
118, 10syl6bbr 280 . . . . . . 7 ((𝜑𝑧𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
1211abbi2dv 2885 . . . . . 6 ((𝜑𝑧𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)})
13 df-rab 3064 . . . . . 6 {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅)}
1412, 13syl6eqr 2817 . . . . 5 ((𝜑𝑧𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
1514ralrimiva 3113 . . . 4 (𝜑 → ∀𝑧𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
16 sneq 4344 . . . . . . 7 (𝑥 = 𝑧 → {𝑥} = {𝑧})
1716fveq2d 6379 . . . . . 6 (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
18 fveq2 6375 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1918ineq1d 3975 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹𝑥) ∩ 𝒫 𝑛) = ((𝐹𝑧) ∩ 𝒫 𝑛))
2019neeq1d 2996 . . . . . . 7 (𝑥 = 𝑧 → (((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅))
2120rabbidv 3338 . . . . . 6 (𝑥 = 𝑧 → {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
2217, 21eqeq12d 2780 . . . . 5 (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅}))
2322cbvralv 3319 . . . 4 (∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∀𝑧𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑧) ∩ 𝒫 𝑛) ≠ ∅})
2415, 23sylibr 225 . . 3 (𝜑 → ∀𝑥𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})
25 toponuni 20998 . . . . . . . . . 10 (𝑗 ∈ (TopOn‘𝑋) → 𝑋 = 𝑗)
26 eqimss2 3818 . . . . . . . . . 10 (𝑋 = 𝑗 𝑗𝑋)
2725, 26syl 17 . . . . . . . . 9 (𝑗 ∈ (TopOn‘𝑋) → 𝑗𝑋)
28 sspwuni 4768 . . . . . . . . 9 (𝑗 ⊆ 𝒫 𝑋 𝑗𝑋)
2927, 28sylibr 225 . . . . . . . 8 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ⊆ 𝒫 𝑋)
3029ad2antlr 718 . . . . . . 7 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 ⊆ 𝒫 𝑋)
31 sseqin2 3979 . . . . . . 7 (𝑗 ⊆ 𝒫 𝑋 ↔ (𝒫 𝑋𝑗) = 𝑗)
3230, 31sylib 209 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = 𝑗)
33 topontop 20997 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3433ad3antlr 722 . . . . . . . . . 10 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑗 ∈ Top)
35 eltop2 21059 . . . . . . . . . 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 4325 . . . . . . . . . . . . . . 15 (𝑜 ∈ 𝒫 𝑋𝑜𝑋)
38 ssralv 3826 . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . . 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 2830 . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . 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 3793 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑜𝑋𝑜 𝑗))
4645biimpa 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜𝑋) → 𝑜 𝑗)
4737, 46sylan2 586 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑜 𝑗)
4847sselda 3761 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥𝑜) → 𝑥 𝑗)
4948adantrr 708 . . . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . . . . . . . 19 𝑗 = 𝑗
5251isneip 21189 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ Top ∧ 𝑥 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ (𝑜 𝑗 ∧ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜))))
5352baibd 535 . . . . . . . . . . . . . . . . 17 (((𝑗 ∈ Top ∧ 𝑥 𝑗) ∧ 𝑜 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ ∃𝑧𝑗 (𝑥𝑧𝑧𝑜)))
5443, 49, 50, 53syl21anc 866 . . . . . . . . . . . . . . . 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 4318 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑜 → 𝒫 𝑛 = 𝒫 𝑜)
5655ineq2d 3976 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑜 → ((𝐹𝑥) ∩ 𝒫 𝑛) = ((𝐹𝑥) ∩ 𝒫 𝑜))
5756neeq1d 2996 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑜 → (((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
5857elrab3 3521 . . . . . . . . . . . . . . . . 17 (𝑜 ∈ 𝒫 𝑋 → (𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
5958ad2antlr 718 . . . . . . . . . . . . . . . 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 300 . . . . . . . . . . . . . . 15 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6160expr 448 . . . . . . . . . . . . . 14 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥𝑜) → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} → (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
6261ralimdva 3109 . . . . . . . . . . . . 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 395 . . . . . . . . . . 11 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → ∀𝑥𝑜 (∃𝑧𝑗 (𝑥𝑧𝑧𝑜) ↔ ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6564an32s 642 . . . . . . . . . 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 3215 . . . . . . . . . 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 270 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (𝑜𝑗 ↔ ∀𝑥𝑜 ((𝐹𝑥) ∩ 𝒫 𝑜) ≠ ∅))
6968rabbi2dva 3981 . . . . . . 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 2817 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋𝑗) = 𝐽)
7132, 70eqtr3d 2801 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽)
7271expl 449 . . . 4 (𝜑 → ((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
7372alrimiv 2022 . . 3 (𝜑 → ∀𝑗((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
74 eleq1 2832 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘𝑋)))
75 fveq2 6375 . . . . . . . 8 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
7675fveq1d 6377 . . . . . . 7 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
7776eqeq1d 2767 . . . . . 6 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
7877ralbidv 3133 . . . . 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 624 . . . 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 3534 . . 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 1494 . 2 (𝜑 → ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
82 df-reu 3062 . 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 225 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 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  ∃!weu 2581  {cab 2751  wne 2937  wral 3055  wrex 3056  ∃!wreu 3057  {crab 3059  cdif 3729  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   cuni 4594  wf 6064  cfv 6068  Topctop 20977  TopOnctopon 20994  neicnei 21181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-topgen 16370  df-top 20978  df-topon 20995  df-nei 21182
This theorem is referenced by: (None)
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