Theorem neibastop3 36893

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	⊢ (𝜑 → 𝑋 ∈ 𝑉)
neibastop1.2	⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))
neibastop1.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
neibastop1.4	⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅}
neibastop1.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣)
neibastop1.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅)

Assertion

Ref	Expression
neibastop3	⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 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.

Step	Hyp	Ref	Expression
1		neibastop1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		neibastop1.2	. . . 4 ⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖ {∅}))
3		neibastop1.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
4		neibastop1.4	. . . 4 ⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅}
5	1, 2, 3, 4	neibastop1 36890	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
6		neibastop1.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣)
7		neibastop1.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅)
8	1, 2, 3, 4, 6, 7	neibastop2 36892	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛 ⊆ 𝑋 ∧ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
9		velpw 4567	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝒫 𝑋 ↔ 𝑛 ⊆ 𝑋)
10	9	anbi1i 635	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅) ↔ (𝑛 ⊆ 𝑋 ∧ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅))
11	8, 10	bitr4di 292	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑧}) ↔ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅)))
12	11	eqabdv 2896	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅)})
13		df-rab 3417	. . . . . 6 ⊢ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∣ (𝑛 ∈ 𝒫 𝑋 ∧ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅)}
14	12, 13	eqtr4di 2816	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅})
15	14	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅})
16		sneq 4599	. . . . . . 7 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
17	16	fveq2d 6885	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧}))
18		fveq2 6881	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
19	18	ineq1d 4172	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ∩ 𝒫 𝑛) = ((𝐹‘𝑧) ∩ 𝒫 𝑛))
20	19	neeq1d 3017	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅))
21	20	rabbidv 3423	. . . . . 6 ⊢ (𝑥 = 𝑧 → {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅})
22	17, 21	eqeq12d 2779	. . . . 5 ⊢ (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅}))
23	22	cbvralvw 3243	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∀𝑧 ∈ 𝑋 ((nei‘𝐽)‘{𝑧}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑧) ∩ 𝒫 𝑛) ≠ ∅})
24	15, 23	sylibr 237	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})
25		toponuni 23071	. . . . . . . . . 10 ⊢ (𝑗 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑗)
26		eqimss2 3996	. . . . . . . . . 10 ⊢ (𝑋 = ∪ 𝑗 → ∪ 𝑗 ⊆ 𝑋)
27	25, 26	syl 18	. . . . . . . . 9 ⊢ (𝑗 ∈ (TopOn‘𝑋) → ∪ 𝑗 ⊆ 𝑋)
28		sspwuni 5066	. . . . . . . . 9 ⊢ (𝑗 ⊆ 𝒫 𝑋 ↔ ∪ 𝑗 ⊆ 𝑋)
29	27, 28	sylibr 237	. . . . . . . 8 ⊢ (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ⊆ 𝒫 𝑋)
30	29	ad2antlr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 ⊆ 𝒫 𝑋)
31		sseqin2 4176	. . . . . . 7 ⊢ (𝑗 ⊆ 𝒫 𝑋 ↔ (𝒫 𝑋 ∩ 𝑗) = 𝑗)
32	30, 31	sylib 221	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋 ∩ 𝑗) = 𝑗)
33		topontop 23070	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
34	33	ad3antlr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑗 ∈ Top)
35		eltop2 23132	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑜 ∈ 𝑗 ↔ ∀𝑥 ∈ 𝑜 ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜)))
36	34, 35	syl 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (𝑜 ∈ 𝑗 ↔ ∀𝑥 ∈ 𝑜 ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜)))
37		elpwi 4569	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ∈ 𝒫 𝑋 → 𝑜 ⊆ 𝑋)
38		ssralv 4006	. . . . . . . . . . . . . . 15 ⊢ (𝑜 ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥 ∈ 𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
39	37, 38	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑜 ∈ 𝒫 𝑋 → (∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥 ∈ 𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
40	39	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥 ∈ 𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
41		simprr 784	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})
42	41	eleq2d 2849	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ 𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
43	33	ad3antlr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑗 ∈ Top)
44	25	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) → 𝑋 = ∪ 𝑗)
45	44	sseq2d 3969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) → (𝑜 ⊆ 𝑋 ↔ 𝑜 ⊆ ∪ 𝑗))
46	45	biimpa 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ⊆ 𝑋) → 𝑜 ⊆ ∪ 𝑗)
47	37, 46	sylan2 604	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → 𝑜 ⊆ ∪ 𝑗)
48	47	sselda 3937	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑜) → 𝑥 ∈ ∪ 𝑗)
49	48	adantrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑥 ∈ ∪ 𝑗)
50	47	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → 𝑜 ⊆ ∪ 𝑗)
51		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝑗 = ∪ 𝑗
52	51	isneip 23262	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ Top ∧ 𝑥 ∈ ∪ 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ (𝑜 ⊆ ∪ 𝑗 ∧ ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜))))
53	52	baibd 548	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 ∈ Top ∧ 𝑥 ∈ ∪ 𝑗) ∧ 𝑜 ⊆ ∪ 𝑗) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜)))
54	43, 49, 50, 53	syl21anc 850	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (𝑜 ∈ ((nei‘𝑗)‘{𝑥}) ↔ ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜)))
55		pweq 4576	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑜 → 𝒫 𝑛 = 𝒫 𝑜)
56	55	ineq2d 4173	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑜 → ((𝐹‘𝑥) ∩ 𝒫 𝑛) = ((𝐹‘𝑥) ∩ 𝒫 𝑜))
57	56	neeq1d 3017	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑜 → (((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅ ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
58	57	elrab3 3651	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 ∈ 𝒫 𝑋 → (𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
59	58	ad2antlr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (𝑜 ∈ {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
60	42, 54, 59	3bitr3d 312	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ (𝑥 ∈ 𝑜 ∧ ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})) → (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
61	60	expr 461	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ 𝑥 ∈ 𝑜) → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} → (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
62	61	ralimdva 3177	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑜 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥 ∈ 𝑜 (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
63	40, 62	syld 48	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} → ∀𝑥 ∈ 𝑜 (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅)))
64	63	imp 411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑜 ∈ 𝒫 𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → ∀𝑥 ∈ 𝑜 (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
65	64	an32s 664	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → ∀𝑥 ∈ 𝑜 (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
66		ralbi 3120	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑜 (∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅) → (∀𝑥 ∈ 𝑜 ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
67	65, 66	syl 18	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑜 ∃𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑜) ↔ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
68	36, 67	bitrd 282	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ 𝑜 ∈ 𝒫 𝑋) → (𝑜 ∈ 𝑗 ↔ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅))
69	68	rabbi2dva 4178	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋 ∩ 𝑗) = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅})
70	69, 4	eqtr4di 2816	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → (𝒫 𝑋 ∩ 𝑗) = 𝐽)
71	32, 70	eqtr3d 2800	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽)
72	71	expl 462	. . . 4 ⊢ (𝜑 → ((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
73	72	alrimiv 1957	. . 3 ⊢ (𝜑 → ∀𝑗((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽))
74		eleq1 2851	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∈ (TopOn‘𝑋) ↔ 𝐽 ∈ (TopOn‘𝑋)))
75		fveq2 6881	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
76	75	fveq1d 6883	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑥}) = ((nei‘𝐽)‘{𝑥}))
77	76	eqeq1d 2765	. . . . . 6 ⊢ (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
78	77	ralbidv 3188	. . . . 5 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∀𝑥 ∈ 𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
79	74, 78	anbi12d 643	. . . 4 ⊢ (𝑗 = 𝐽 → ((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ↔ (𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})))
80	79	eqeu 3669	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝐽)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) ∧ ∀𝑗((𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}) → 𝑗 = 𝐽)) → ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
81	5, 5, 24, 73, 80	syl121anc 1402	. 2 ⊢ (𝜑 → ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
82		df-reu 3370	. 2 ⊢ (∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅} ↔ ∃!𝑗(𝑗 ∈ (TopOn‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅}))
83	81, 82	sylibr 237	1 ⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑥 ∈ 𝑋 ((nei‘𝑗)‘{𝑥}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((𝐹‘𝑥) ∩ 𝒫 𝑛) ≠ ∅})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃!weu 2596  {cab 2741   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367  {crab 3416   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872  ⟶wf 6532  ‘cfv 6536  Topctop 23050  TopOnctopon 23067  neicnei 23254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-topgen 17491  df-top 23051  df-topon 23068  df-nei 23255

This theorem is referenced by: (None)