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Theorem neipcfilu 24278

Description: In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)

**Hypotheses**
Ref	Expression
neipcfilu.x	⊢ 𝑋 = (Base‘𝑊)
neipcfilu.j	⊢ 𝐽 = (TopOpen‘𝑊)
neipcfilu.u	⊢ 𝑈 = (UnifSt‘𝑊)

**Assertion**
Ref	Expression
neipcfilu	⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFil_u‘𝑈))

Proof of Theorem neipcfilu Dummy variables 𝑣 𝑎 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1143	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑊 ∈ TopSp)
2		neipcfilu.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑊)
3		neipcfilu.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
4	2, 3	istps 22917	. . . . 5 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
5	1, 4	sylib 219	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6		simp3 1144	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
7	6	snssd 4718	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ⊆ 𝑋)
8	6	snn0d 4707	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ≠ ∅)
9		neifil 23863	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
10	5, 7, 8, 9	syl3anc 1379	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
11		filfbas 23831	. . 3 ⊢ (((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋))
12	10, 11	syl 17	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋))
13		eqid 2739	. . . . . . . . . 10 ⊢ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
14		imaeq1 6007	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
15	14	rspceeqv 3583	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
16	13, 15	mpan2 697	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
17		vex 3435	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18	17	imaex 7854	. . . . . . . . . 10 ⊢ (𝑤 “ {𝑃}) ∈ V
19		eqid 2739	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
20	19	elrnmpt 5900	. . . . . . . . . 10 ⊢ ((𝑤 “ {𝑃}) ∈ V → ((𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})))
21	18, 20	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
22	16, 21	sylibr 235	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
23	22	ad2antlr 733	. . . . . . 7 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
24		neipcfilu.u	. . . . . . . . . . . . 13 ⊢ 𝑈 = (UnifSt‘𝑊)
25	2, 24, 3	isusp 24244	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
26	25	simplbi 497	. . . . . . . . . . 11 ⊢ (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
27	26	3ad2ant1 1139	. . . . . . . . . 10 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
28		eqid 2739	. . . . . . . . . . 11 ⊢ (unifTop‘𝑈) = (unifTop‘𝑈)
29	28	utopsnneip 24231	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
30	27, 6, 29	syl2anc 590	. . . . . . . . 9 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
31	30	eleq2d 2825	. . . . . . . 8 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
32	31	ad3antrrr 736	. . . . . . 7 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
33	23, 32	mpbird 258	. . . . . 6 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}))
34		simpl1 1198	. . . . . . . . . 10 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ (𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
35	34	3anassrs 1367	. . . . . . . . 9 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
36	25	simprbi 498	. . . . . . . . 9 ⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘𝑈))
37	35, 36	syl 17	. . . . . . . 8 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝐽 = (unifTop‘𝑈))
38	37	fveq2d 6831	. . . . . . 7 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
39	38	fveq1d 6829	. . . . . 6 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃}))
40	33, 39	eleqtrrd 2842	. . . . 5 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
41		simpr 485	. . . . 5 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
42		id 22	. . . . . . . 8 ⊢ (𝑎 = (𝑤 “ {𝑃}) → 𝑎 = (𝑤 “ {𝑃}))
43	42	sqxpeqd 5650	. . . . . . 7 ⊢ (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
44	43	sseq1d 3946	. . . . . 6 ⊢ (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
45	44	rspcev 3560	. . . . 5 ⊢ (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
46	40, 41, 45	syl2anc 590	. . . 4 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
47	27	adantr 481	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
48	6	adantr 481	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑃 ∈ 𝑋)
49		simpr 485	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑈)
50		simpll1 1219	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
51		simplr 774	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝑈)
52		ustexsym 24199	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))
53	50, 51, 52	syl2anc 590	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))
54	50	ad2antrr 732	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
55		simplr 774	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ∈ 𝑈)
56		ustssxp 24188	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
57	54, 55, 56	syl2anc 590	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
58		simpll2 1220	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ ((𝑢 ∘ 𝑢) ⊆ 𝑣 ∧ 𝑤 ∈ 𝑈 ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))) → 𝑃 ∈ 𝑋)
59	58	3anassrs 1367	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑃 ∈ 𝑋)
60		ustneism 24207	. . . . . . . . . . 11 ⊢ ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤 ∘ ^◡𝑤))
61	57, 59, 60	syl2anc 590	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤 ∘ ^◡𝑤))
62		simprl 776	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ^◡𝑤 = 𝑤)
63	62	coeq2d 5804	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ ^◡𝑤) = (𝑤 ∘ 𝑤))
64		coss1 5797	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑤))
65		coss2 5798	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑢 → (𝑢 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
66	64, 65	sstrd 3925	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
67	66	ad2antll 735	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
68		simpllr 781	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑢 ∘ 𝑢) ⊆ 𝑣)
69	67, 68	sstrd 3925	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
70	63, 69	eqsstrd 3949	. . . . . . . . . 10 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ ^◡𝑤) ⊆ 𝑣)
71	61, 70	sstrd 3925	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
72	71	ex 413	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) → ((^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
73	72	reximdva 3152	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → (∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
74	53, 73	mpd 15	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
75		ustexhalf 24194	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣)
76	75	3adant2 1137	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣)
77	74, 76	r19.29a 3147	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
78	47, 48, 49, 77	syl3anc 1379	. . . 4 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
79	46, 78	r19.29a 3147	. . 3 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
80	79	ralrimiva 3131	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
81		iscfilu 24270	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFil_u‘𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
82	27, 81	syl 17	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFil_u‘𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
83	12, 80, 82	mpbir2and 719	1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFil_u‘𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ⊆ wss 3883 ∅c0 4261 {csn 4555 ↦ cmpt 5153 × cxp 5616 ^◡ccnv 5617 ran crn 5619 “ cima 5621 ∘ ccom 5622 ‘cfv 6485 Basecbs 17170 TopOpenctopn 17375 fBascfbas 21335 TopOnctopon 22893 TopSpctps 22915 neicnei 23080 Filcfil 23828 UnifOncust 24183 unifTopcutop 24213 UnifStcuss 24236 UnifSpcusp 24237 CauFil_uccfilu 24268

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-om 7807 df-1o 8395 df-2o 8396 df-en 8884 df-fin 8887 df-fi 9314 df-fbas 21344 df-top 22877 df-topon 22894 df-topsp 22916 df-nei 23081 df-fil 23829 df-ust 24184 df-utop 24214 df-usp 24240 df-cfilu 24269

This theorem is referenced by: ucnextcn 24286

W3C validator