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Theorem neipcfilu 24181

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 1137	. . . . 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 22819	. . . . 5 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
5	1, 4	sylib 218	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6		simp3 1138	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
7	6	snssd 4760	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ⊆ 𝑋)
8	6	snn0d 4727	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ≠ ∅)
9		neifil 23765	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
10	5, 7, 8, 9	syl3anc 1373	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
11		filfbas 23733	. . 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 2729	. . . . . . . . . 10 ⊢ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
14		imaeq1 6006	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
15	14	rspceeqv 3600	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
16	13, 15	mpan2 691	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
17		vex 3440	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18	17	imaex 7847	. . . . . . . . . 10 ⊢ (𝑤 “ {𝑃}) ∈ V
19		eqid 2729	. . . . . . . . . . 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 234	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
23	22	ad2antlr 727	. . . . . . 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 24147	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
26	25	simplbi 497	. . . . . . . . . . 11 ⊢ (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
27	26	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
28		eqid 2729	. . . . . . . . . . 11 ⊢ (unifTop‘𝑈) = (unifTop‘𝑈)
29	28	utopsnneip 24134	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
30	27, 6, 29	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
31	30	eleq2d 2814	. . . . . . . 8 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
32	31	ad3antrrr 730	. . . . . . 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 257	. . . . . 6 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}))
34		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ (𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
35	34	3anassrs 1361	. . . . . . . . 9 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
36	25	simprbi 496	. . . . . . . . 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 6826	. . . . . . 7 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
39	38	fveq1d 6824	. . . . . 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 2831	. . . . 5 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
41		simpr 484	. . . . 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 5651	. . . . . . 7 ⊢ (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
44	43	sseq1d 3967	. . . . . 6 ⊢ (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
45	44	rspcev 3577	. . . . 5 ⊢ (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
46	40, 41, 45	syl2anc 584	. . . 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 480	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
48	6	adantr 480	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑃 ∈ 𝑋)
49		simpr 484	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑈)
50		simpll1 1213	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
51		simplr 768	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝑈)
52		ustexsym 24101	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))
53	50, 51, 52	syl2anc 584	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))
54	50	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
55		simplr 768	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ∈ 𝑈)
56		ustssxp 24090	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
57	54, 55, 56	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
58		simpll2 1214	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ ((𝑢 ∘ 𝑢) ⊆ 𝑣 ∧ 𝑤 ∈ 𝑈 ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))) → 𝑃 ∈ 𝑋)
59	58	3anassrs 1361	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑃 ∈ 𝑋)
60		ustneism 24109	. . . . . . . . . . 11 ⊢ ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤 ∘ ^◡𝑤))
61	57, 59, 60	syl2anc 584	. . . . . . . . . 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 770	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ^◡𝑤 = 𝑤)
63	62	coeq2d 5805	. . . . . . . . . . 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 5798	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑤))
65		coss2 5799	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑢 → (𝑢 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
66	64, 65	sstrd 3946	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
67	66	ad2antll 729	. . . . . . . . . . . 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 775	. . . . . . . . . . . 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 3946	. . . . . . . . . . 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 3970	. . . . . . . . . 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 3946	. . . . . . . . 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 412	. . . . . . . 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 3142	. . . . . . 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 24096	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣)
76	75	3adant2 1131	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣)
77	74, 76	r19.29a 3137	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
78	47, 48, 49, 77	syl3anc 1373	. . . 4 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
79	46, 78	r19.29a 3137	. . 3 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
80	79	ralrimiva 3121	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
81		iscfilu 24173	. . 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 713	1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFil_u‘𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 Vcvv 3436 ⊆ wss 3903 ∅c0 4284 {csn 4577 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 ran crn 5620 “ cima 5622 ∘ ccom 5623 ‘cfv 6482 Basecbs 17120 TopOpenctopn 17325 fBascfbas 21249 TopOnctopon 22795 TopSpctps 22817 neicnei 22982 Filcfil 23730 UnifOncust 24085 unifTopcutop 24116 UnifStcuss 24139 UnifSpcusp 24140 CauFil_uccfilu 24171

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-om 7800 df-1o 8388 df-2o 8389 df-en 8873 df-fin 8876 df-fi 9301 df-fbas 21258 df-top 22779 df-topon 22796 df-topsp 22818 df-nei 22983 df-fil 23731 df-ust 24086 df-utop 24117 df-usp 24143 df-cfilu 24172

This theorem is referenced by: ucnextcn 24189

W3C validator