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Theorem neipcfilu 24343

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 1149	. . . . 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 22982	. . . . 5 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
5	1, 4	sylib 220	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6		simp3 1150	. . . . 5 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
7	6	snssd 4742	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ⊆ 𝑋)
8	6	snn0d 4731	. . . 4 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ≠ ∅)
9		neifil 23928	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
10	5, 7, 8, 9	syl3anc 1389	. . 3 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
11		filfbas 23896	. . 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 2761	. . . . . . . . . 10 ⊢ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
14		imaeq1 6040	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
15	14	rspceeqv 3603	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
16	13, 15	mpan2 701	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
17		vex 3457	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
18	17	imaex 7890	. . . . . . . . . 10 ⊢ (𝑤 “ {𝑃}) ∈ V
19		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))
20	19	elrnmpt 5930	. . . . . . . . . 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 236	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
23	22	ad2antlr 737	. . . . . . 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 24309	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
26	25	simplbi 500	. . . . . . . . . . 11 ⊢ (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
27	26	3ad2ant1 1145	. . . . . . . . . 10 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
28		eqid 2761	. . . . . . . . . . 11 ⊢ (unifTop‘𝑈) = (unifTop‘𝑈)
29	28	utopsnneip 24296	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
30	27, 6, 29	syl2anc 593	. . . . . . . . 9 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))
31	30	eleq2d 2847	. . . . . . . 8 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))))
32	31	ad3antrrr 740	. . . . . . 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 259	. . . . . 6 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}))
34		simpl1 1204	. . . . . . . . . 10 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ (𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
35	34	3anassrs 1373	. . . . . . . . 9 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
36	25	simprbi 501	. . . . . . . . 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 6866	. . . . . . 7 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
39	38	fveq1d 6864	. . . . . 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 2864	. . . . 5 ⊢ (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
41		simpr 488	. . . . 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 5675	. . . . . . 7 ⊢ (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
44	43	sseq1d 3965	. . . . . 6 ⊢ (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
45	44	rspcev 3580	. . . . 5 ⊢ (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
46	40, 41, 45	syl2anc 593	. . . 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 484	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
48	6	adantr 484	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑃 ∈ 𝑋)
49		simpr 488	. . . . 5 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑈)
50		simpll1 1225	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
51		simplr 778	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝑈)
52		ustexsym 24264	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))
53	50, 51, 52	syl2anc 593	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))
54	50	ad2antrr 736	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
55		simplr 778	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ∈ 𝑈)
56		ustssxp 24253	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
57	54, 55, 56	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
58		simpll2 1226	. . . . . . . . . . . 12 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ ((𝑢 ∘ 𝑢) ⊆ 𝑣 ∧ 𝑤 ∈ 𝑈 ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))) → 𝑃 ∈ 𝑋)
59	58	3anassrs 1373	. . . . . . . . . . 11 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑃 ∈ 𝑋)
60		ustneism 24272	. . . . . . . . . . 11 ⊢ ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤 ∘ ^◡𝑤))
61	57, 59, 60	syl2anc 593	. . . . . . . . . 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 780	. . . . . . . . . . . 12 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ^◡𝑤 = 𝑤)
63	62	coeq2d 5830	. . . . . . . . . . 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 5823	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑤))
65		coss2 5824	. . . . . . . . . . . . . 14 ⊢ (𝑤 ⊆ 𝑢 → (𝑢 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
66	64, 65	sstrd 3944	. . . . . . . . . . . . 13 ⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢))
67	66	ad2antll 739	. . . . . . . . . . . 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 785	. . . . . . . . . . . 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 3944	. . . . . . . . . . 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 3968	. . . . . . . . . 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 3944	. . . . . . . . 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 416	. . . . . . . 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 3174	. . . . . . 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 24259	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣)
76	75	3adant2 1143	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣)
77	74, 76	r19.29a 3169	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
78	47, 48, 49, 77	syl3anc 1389	. . . 4 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
79	46, 78	r19.29a 3169	. . 3 ⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
80	79	ralrimiva 3153	. 2 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
81		iscfilu 24335	. . 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 723	1 ⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFil_u‘𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ⊆ wss 3902 ∅c0 4283 {csn 4579 ↦ cmpt 5178 × cxp 5641 ^◡ccnv 5642 ran crn 5644 “ cima 5646 ∘ ccom 5647 ‘cfv 6516 Basecbs 17236 TopOpenctopn 17441 fBascfbas 21400 TopOnctopon 22958 TopSpctps 22980 neicnei 23145 Filcfil 23893 UnifOncust 24248 unifTopcutop 24278 UnifStcuss 24301 UnifSpcusp 24302 CauFil_uccfilu 24333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-om 7842 df-1o 8431 df-2o 8432 df-en 8922 df-fin 8925 df-fi 9351 df-fbas 21409 df-top 22942 df-topon 22959 df-topsp 22981 df-nei 23146 df-fil 23894 df-ust 24249 df-utop 24279 df-usp 24305 df-cfilu 24334

This theorem is referenced by: ucnextcn 24351

W3C validator