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Theorem neipcfilu 22832
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 ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))

Proof of Theorem neipcfilu
Dummy variables 𝑣 𝑎 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1129 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑊 ∈ TopSp)
2 neipcfilu.x . . . . . 6 𝑋 = (Base‘𝑊)
3 neipcfilu.j . . . . . 6 𝐽 = (TopOpen‘𝑊)
42, 3istps 21470 . . . . 5 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 219 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6 simp3 1130 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑃𝑋)
76snssd 4734 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ⊆ 𝑋)
8 snnzg 4702 . . . . 5 (𝑃𝑋 → {𝑃} ≠ ∅)
96, 8syl 17 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ≠ ∅)
10 neifil 22416 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
115, 7, 9, 10syl3anc 1363 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
12 filfbas 22384 . . 3 (((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋))
1311, 12syl 17 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋))
14 eqid 2818 . . . . . . . . . 10 (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
15 imaeq1 5917 . . . . . . . . . . 11 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
1615rspceeqv 3635 . . . . . . . . . 10 ((𝑤𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
1714, 16mpan2 687 . . . . . . . . 9 (𝑤𝑈 → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
18 vex 3495 . . . . . . . . . . 11 𝑤 ∈ V
1918imaex 7610 . . . . . . . . . 10 (𝑤 “ {𝑃}) ∈ V
20 eqid 2818 . . . . . . . . . . 11 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
2120elrnmpt 5821 . . . . . . . . . 10 ((𝑤 “ {𝑃}) ∈ V → ((𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})))
2219, 21ax-mp 5 . . . . . . . . 9 ((𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
2317, 22sylibr 235 . . . . . . . 8 (𝑤𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2423ad2antlr 723 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
25 neipcfilu.u . . . . . . . . . . . . 13 𝑈 = (UnifSt‘𝑊)
262, 25, 3isusp 22797 . . . . . . . . . . . 12 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
2726simplbi 498 . . . . . . . . . . 11 (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
28273ad2ant1 1125 . . . . . . . . . 10 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
29 eqid 2818 . . . . . . . . . . 11 (unifTop‘𝑈) = (unifTop‘𝑈)
3029utopsnneip 22784 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3128, 6, 30syl2anc 584 . . . . . . . . 9 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3231eleq2d 2895 . . . . . . . 8 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
3332ad3antrrr 726 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
3424, 33mpbird 258 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}))
35 simpl1 1183 . . . . . . . . . 10 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ (𝑣𝑈𝑤𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
36353anassrs 1352 . . . . . . . . 9 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
3726simprbi 497 . . . . . . . . 9 (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘𝑈))
3836, 37syl 17 . . . . . . . 8 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝐽 = (unifTop‘𝑈))
3938fveq2d 6667 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
4039fveq1d 6665 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃}))
4134, 40eleqtrrd 2913 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
42 simpr 485 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
43 id 22 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑃}) → 𝑎 = (𝑤 “ {𝑃}))
4443sqxpeqd 5580 . . . . . . 7 (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
4544sseq1d 3995 . . . . . 6 (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
4645rspcev 3620 . . . . 5 (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4741, 42, 46syl2anc 584 . . . 4 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4828adantr 481 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
496adantr 481 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑃𝑋)
50 simpr 485 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑣𝑈)
51 simpll1 1204 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
52 simplr 765 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑢𝑈)
53 ustexsym 22751 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5451, 52, 53syl2anc 584 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5551ad2antrr 722 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
56 simplr 765 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤𝑈)
57 ustssxp 22740 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5855, 56, 57syl2anc 584 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
59 simpll2 1205 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ ((𝑢𝑢) ⊆ 𝑣𝑤𝑈 ∧ (𝑤 = 𝑤𝑤𝑢))) → 𝑃𝑋)
60593anassrs 1352 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑃𝑋)
61 ustneism 22759 . . . . . . . . . . 11 ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
6258, 60, 61syl2anc 584 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
63 simprl 767 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 = 𝑤)
6463coeq2d 5726 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) = (𝑤𝑤))
65 coss1 5719 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑤))
66 coss2 5720 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑢𝑤) ⊆ (𝑢𝑢))
6765, 66sstrd 3974 . . . . . . . . . . . . 13 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑢))
6867ad2antll 725 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ (𝑢𝑢))
69 simpllr 772 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑢𝑢) ⊆ 𝑣)
7068, 69sstrd 3974 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7164, 70eqsstrd 4002 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7262, 71sstrd 3974 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7372ex 413 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) → ((𝑤 = 𝑤𝑤𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7473reximdva 3271 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → (∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7554, 74mpd 15 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
76 ustexhalf 22746 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
77763adant2 1123 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
7875, 77r19.29a 3286 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7948, 49, 50, 78syl3anc 1363 . . . 4 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
8047, 79r19.29a 3286 . . 3 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
8180ralrimiva 3179 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
82 iscfilu 22824 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
8328, 82syl 17 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
8413, 81, 83mpbir2and 709 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288  {csn 4557  cmpt 5137   × cxp 5546  ccnv 5547  ran crn 5549  cima 5551  ccom 5552  cfv 6348  Basecbs 16471  TopOpenctopn 16683  fBascfbas 20461  TopOnctopon 21446  TopSpctps 21468  neicnei 21633  Filcfil 22381  UnifOncust 22735  unifTopcutop 22766  UnifStcuss 22789  UnifSpcusp 22790  CauFiluccfilu 22822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-fin 8501  df-fi 8863  df-fbas 20470  df-top 21430  df-topon 21447  df-topsp 21469  df-nei 21634  df-fil 22382  df-ust 22736  df-utop 22767  df-usp 22793  df-cfilu 22823
This theorem is referenced by:  ucnextcn  22840
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