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Theorem neipcfilu 23599
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 1137 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑊 ∈ TopSp)
2 neipcfilu.x . . . . . 6 𝑋 = (Base‘𝑊)
3 neipcfilu.j . . . . . 6 𝐽 = (TopOpen‘𝑊)
42, 3istps 22234 . . . . 5 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 217 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6 simp3 1138 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑃𝑋)
76snssd 4767 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ⊆ 𝑋)
86snn0d 4734 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ≠ ∅)
9 neifil 23182 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
105, 7, 8, 9syl3anc 1371 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
11 filfbas 23150 . . 3 (((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋))
1210, 11syl 17 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋))
13 eqid 2737 . . . . . . . . . 10 (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
14 imaeq1 6006 . . . . . . . . . . 11 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
1514rspceeqv 3593 . . . . . . . . . 10 ((𝑤𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
1613, 15mpan2 689 . . . . . . . . 9 (𝑤𝑈 → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
17 vex 3447 . . . . . . . . . . 11 𝑤 ∈ V
1817imaex 7845 . . . . . . . . . 10 (𝑤 “ {𝑃}) ∈ V
19 eqid 2737 . . . . . . . . . . 11 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
2019elrnmpt 5909 . . . . . . . . . 10 ((𝑤 “ {𝑃}) ∈ V → ((𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})))
2118, 20ax-mp 5 . . . . . . . . 9 ((𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
2216, 21sylibr 233 . . . . . . . 8 (𝑤𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2322ad2antlr 725 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
24 neipcfilu.u . . . . . . . . . . . . 13 𝑈 = (UnifSt‘𝑊)
252, 24, 3isusp 23564 . . . . . . . . . . . 12 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
2625simplbi 498 . . . . . . . . . . 11 (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
27263ad2ant1 1133 . . . . . . . . . 10 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
28 eqid 2737 . . . . . . . . . . 11 (unifTop‘𝑈) = (unifTop‘𝑈)
2928utopsnneip 23551 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3027, 6, 29syl2anc 584 . . . . . . . . 9 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3130eleq2d 2823 . . . . . . . 8 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
3231ad3antrrr 728 . . . . . . 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}))))
3323, 32mpbird 256 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}))
34 simpl1 1191 . . . . . . . . . 10 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ (𝑣𝑈𝑤𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
35343anassrs 1360 . . . . . . . . 9 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
3625simprbi 497 . . . . . . . . 9 (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘𝑈))
3735, 36syl 17 . . . . . . . 8 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝐽 = (unifTop‘𝑈))
3837fveq2d 6843 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
3938fveq1d 6841 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃}))
4033, 39eleqtrrd 2841 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
41 simpr 485 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
42 id 22 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑃}) → 𝑎 = (𝑤 “ {𝑃}))
4342sqxpeqd 5663 . . . . . . 7 (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
4443sseq1d 3973 . . . . . 6 (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
4544rspcev 3579 . . . . 5 (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4640, 41, 45syl2anc 584 . . . 4 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4727adantr 481 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
486adantr 481 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑃𝑋)
49 simpr 485 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑣𝑈)
50 simpll1 1212 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
51 simplr 767 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑢𝑈)
52 ustexsym 23518 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5350, 51, 52syl2anc 584 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5450ad2antrr 724 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
55 simplr 767 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤𝑈)
56 ustssxp 23507 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5754, 55, 56syl2anc 584 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
58 simpll2 1213 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ ((𝑢𝑢) ⊆ 𝑣𝑤𝑈 ∧ (𝑤 = 𝑤𝑤𝑢))) → 𝑃𝑋)
59583anassrs 1360 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑃𝑋)
60 ustneism 23526 . . . . . . . . . . 11 ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
6157, 59, 60syl2anc 584 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
62 simprl 769 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 = 𝑤)
6362coeq2d 5816 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) = (𝑤𝑤))
64 coss1 5809 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑤))
65 coss2 5810 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑢𝑤) ⊆ (𝑢𝑢))
6664, 65sstrd 3952 . . . . . . . . . . . . 13 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑢))
6766ad2antll 727 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ (𝑢𝑢))
68 simpllr 774 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑢𝑢) ⊆ 𝑣)
6967, 68sstrd 3952 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7063, 69eqsstrd 3980 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7161, 70sstrd 3952 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7271ex 413 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) → ((𝑤 = 𝑤𝑤𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7372reximdva 3163 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → (∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7453, 73mpd 15 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
75 ustexhalf 23513 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
76753adant2 1131 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
7774, 76r19.29a 3157 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7847, 48, 49, 77syl3anc 1371 . . . 4 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7946, 78r19.29a 3157 . . 3 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
8079ralrimiva 3141 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
81 iscfilu 23591 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
8227, 81syl 17 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
8312, 80, 82mpbir2and 711 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  Vcvv 3443  wss 3908  c0 4280  {csn 4584  cmpt 5186   × cxp 5629  ccnv 5630  ran crn 5632  cima 5634  ccom 5635  cfv 6493  Basecbs 17042  TopOpenctopn 17262  fBascfbas 20736  TopOnctopon 22210  TopSpctps 22232  neicnei 22399  Filcfil 23147  UnifOncust 23502  unifTopcutop 23533  UnifStcuss 23556  UnifSpcusp 23557  CauFiluccfilu 23589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-om 7795  df-1o 8404  df-er 8606  df-en 8842  df-fin 8845  df-fi 9305  df-fbas 20745  df-top 22194  df-topon 22211  df-topsp 22233  df-nei 22400  df-fil 23148  df-ust 23503  df-utop 23534  df-usp 23560  df-cfilu 23590
This theorem is referenced by:  ucnextcn  23607
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