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Theorem neipcfilu 22147
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 1082 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑊 ∈ TopSp)
2 neipcfilu.x . . . . . 6 𝑋 = (Base‘𝑊)
3 neipcfilu.j . . . . . 6 𝐽 = (TopOpen‘𝑊)
42, 3istps 20786 . . . . 5 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 208 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6 simp3 1083 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑃𝑋)
76snssd 4372 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ⊆ 𝑋)
8 snnzg 4339 . . . . 5 (𝑃𝑋 → {𝑃} ≠ ∅)
96, 8syl 17 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ≠ ∅)
10 neifil 21731 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
115, 7, 9, 10syl3anc 1366 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
12 filfbas 21699 . . 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 2651 . . . . . . . . . 10 (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
15 imaeq1 5496 . . . . . . . . . . . 12 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
1615eqeq2d 2661 . . . . . . . . . . 11 (𝑣 = 𝑤 → ((𝑤 “ {𝑃}) = (𝑣 “ {𝑃}) ↔ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})))
1716rspcev 3340 . . . . . . . . . 10 ((𝑤𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
1814, 17mpan2 707 . . . . . . . . 9 (𝑤𝑈 → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
19 vex 3234 . . . . . . . . . . 11 𝑤 ∈ V
2019imaex 7146 . . . . . . . . . 10 (𝑤 “ {𝑃}) ∈ V
21 eqid 2651 . . . . . . . . . . 11 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
2221elrnmpt 5404 . . . . . . . . . 10 ((𝑤 “ {𝑃}) ∈ V → ((𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})))
2320, 22ax-mp 5 . . . . . . . . 9 ((𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
2418, 23sylibr 224 . . . . . . . 8 (𝑤𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
2524ad2antlr 763 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
26 neipcfilu.u . . . . . . . . . . . . 13 𝑈 = (UnifSt‘𝑊)
272, 26, 3isusp 22112 . . . . . . . . . . . 12 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
2827simplbi 475 . . . . . . . . . . 11 (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
29283ad2ant1 1102 . . . . . . . . . 10 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
30 eqid 2651 . . . . . . . . . . 11 (unifTop‘𝑈) = (unifTop‘𝑈)
3130utopsnneip 22099 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3229, 6, 31syl2anc 694 . . . . . . . . 9 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3332eleq2d 2716 . . . . . . . 8 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
3433ad3antrrr 766 . . . . . . 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}))))
3525, 34mpbird 247 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}))
36 simpl1 1084 . . . . . . . . . 10 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ (𝑣𝑈𝑤𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
37363anassrs 1313 . . . . . . . . 9 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
3827simprbi 479 . . . . . . . . 9 (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘𝑈))
3937, 38syl 17 . . . . . . . 8 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝐽 = (unifTop‘𝑈))
4039fveq2d 6233 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
4140fveq1d 6231 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃}))
4235, 41eleqtrrd 2733 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
43 simpr 476 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
44 id 22 . . . . . . . 8 (𝑎 = (𝑤 “ {𝑃}) → 𝑎 = (𝑤 “ {𝑃}))
4544sqxpeqd 5175 . . . . . . 7 (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
4645sseq1d 3665 . . . . . 6 (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
4746rspcev 3340 . . . . 5 (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4842, 43, 47syl2anc 694 . . . 4 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4929adantr 480 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
506adantr 480 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑃𝑋)
51 simpr 476 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑣𝑈)
52 simpll1 1120 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
53 simplr 807 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑢𝑈)
54 ustexsym 22066 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5552, 53, 54syl2anc 694 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5652ad2antrr 762 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
57 simplr 807 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤𝑈)
58 ustssxp 22055 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5956, 57, 58syl2anc 694 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
60 simpll2 1121 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ ((𝑢𝑢) ⊆ 𝑣𝑤𝑈 ∧ (𝑤 = 𝑤𝑤𝑢))) → 𝑃𝑋)
61603anassrs 1313 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑃𝑋)
62 ustneism 22074 . . . . . . . . . . 11 ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
6359, 61, 62syl2anc 694 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
64 simprl 809 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 = 𝑤)
6564coeq2d 5317 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) = (𝑤𝑤))
66 coss1 5310 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑤))
67 coss2 5311 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑢𝑤) ⊆ (𝑢𝑢))
6866, 67sstrd 3646 . . . . . . . . . . . . 13 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑢))
6968ad2antll 765 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ (𝑢𝑢))
70 simpllr 815 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑢𝑢) ⊆ 𝑣)
7169, 70sstrd 3646 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7265, 71eqsstrd 3672 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7363, 72sstrd 3646 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7473ex 449 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) → ((𝑤 = 𝑤𝑤𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7574reximdva 3046 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → (∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7655, 75mpd 15 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
77 ustexhalf 22061 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
78773adant2 1100 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
7976, 78r19.29a 3107 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
8049, 50, 51, 79syl3anc 1366 . . . 4 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
8148, 80r19.29a 3107 . . 3 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
8281ralrimiva 2995 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
83 iscfilu 22139 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
8429, 83syl 17 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)))
8513, 82, 84mpbir2and 977 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  {csn 4210  cmpt 4762   × cxp 5141  ccnv 5142  ran crn 5144  cima 5146  ccom 5147  cfv 5926  Basecbs 15904  TopOpenctopn 16129  fBascfbas 19782  TopOnctopon 20763  TopSpctps 20784  neicnei 20949  Filcfil 21696  UnifOncust 22050  unifTopcutop 22081  UnifStcuss 22104  UnifSpcusp 22105  CauFiluccfilu 22137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-fbas 19791  df-top 20747  df-topon 20764  df-topsp 20785  df-nei 20950  df-fil 21697  df-ust 22051  df-utop 22082  df-usp 22108  df-cfilu 22138
This theorem is referenced by:  ucnextcn  22155
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