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Theorem neipcfilu 22010
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 1060 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑊 ∈ TopSp)
2 neipcfilu.x . . . . . 6 𝑋 = (Base‘𝑊)
3 neipcfilu.j . . . . . 6 𝐽 = (TopOpen‘𝑊)
42, 3istps 20651 . . . . 5 (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 208 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6 simp3 1061 . . . . 5 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑃𝑋)
76snssd 4309 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ⊆ 𝑋)
8 snnzg 4278 . . . . 5 (𝑃𝑋 → {𝑃} ≠ ∅)
96, 8syl 17 . . . 4 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → {𝑃} ≠ ∅)
10 neifil 21594 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
115, 7, 9, 10syl3anc 1323 . . 3 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋))
12 filfbas 21562 . . 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 2621 . . . . . . . . . 10 (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})
15 imaeq1 5420 . . . . . . . . . . . 12 (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃}))
1615eqeq2d 2631 . . . . . . . . . . 11 (𝑣 = 𝑤 → ((𝑤 “ {𝑃}) = (𝑣 “ {𝑃}) ↔ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})))
1716rspcev 3295 . . . . . . . . . 10 ((𝑤𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
1814, 17mpan2 706 . . . . . . . . 9 (𝑤𝑈 → ∃𝑣𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))
19 vex 3189 . . . . . . . . . . 11 𝑤 ∈ V
2019imaex 7051 . . . . . . . . . 10 (𝑤 “ {𝑃}) ∈ V
21 eqid 2621 . . . . . . . . . . 11 (𝑣𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣𝑈 ↦ (𝑣 “ {𝑃}))
2221elrnmpt 5332 . . . . . . . . . 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 762 . . . . . . 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 21975 . . . . . . . . . . . 12 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈)))
2827simplbi 476 . . . . . . . . . . 11 (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋))
29283ad2ant1 1080 . . . . . . . . . 10 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
30 eqid 2621 . . . . . . . . . . 11 (unifTop‘𝑈) = (unifTop‘𝑈)
3130utopsnneip 21962 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3229, 6, 31syl2anc 692 . . . . . . . . 9 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑃})))
3332eleq2d 2684 . . . . . . . 8 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑃}))))
3433ad3antrrr 765 . . . . . . 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 1062 . . . . . . . . . 10 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ (𝑣𝑈𝑤𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp)
37363anassrs 1287 . . . . . . . . 9 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp)
3827simprbi 480 . . . . . . . . 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 6152 . . . . . . 7 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈)))
4140fveq1d 6150 . . . . . 6 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃}))
4235, 41eleqtrrd 2701 . . . . 5 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}))
43 simpr 477 . . . . 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 5101 . . . . . . 7 (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})))
4645sseq1d 3611 . . . . . 6 (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
4746rspcev 3295 . . . . 5 (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4842, 43, 47syl2anc 692 . . . 4 (((((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) ∧ 𝑤𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
4929adantr 481 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
506adantr 481 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑃𝑋)
51 simpr 477 . . . . 5 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → 𝑣𝑈)
52 simpll1 1098 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋))
53 simplr 791 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → 𝑢𝑈)
54 ustexsym 21929 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5552, 53, 54syl2anc 692 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) → ∃𝑤𝑈 (𝑤 = 𝑤𝑤𝑢))
5652ad2antrr 761 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑈 ∈ (UnifOn‘𝑋))
57 simplr 791 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤𝑈)
58 ustssxp 21918 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5956, 57, 58syl2anc 692 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋))
60 simpll2 1099 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ ((𝑢𝑢) ⊆ 𝑣𝑤𝑈 ∧ (𝑤 = 𝑤𝑤𝑢))) → 𝑃𝑋)
61603anassrs 1287 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑃𝑋)
62 ustneism 21937 . . . . . . . . . . 11 ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
6359, 61, 62syl2anc 692 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤𝑤))
64 simprl 793 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → 𝑤 = 𝑤)
6564coeq2d 5244 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) = (𝑤𝑤))
66 coss1 5237 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑤))
67 coss2 5238 . . . . . . . . . . . . . 14 (𝑤𝑢 → (𝑢𝑤) ⊆ (𝑢𝑢))
6866, 67sstrd 3593 . . . . . . . . . . . . 13 (𝑤𝑢 → (𝑤𝑤) ⊆ (𝑢𝑢))
6968ad2antll 764 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ (𝑢𝑢))
70 simpllr 798 . . . . . . . . . . . 12 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑢𝑢) ⊆ 𝑣)
7169, 70sstrd 3593 . . . . . . . . . . 11 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7265, 71eqsstrd 3618 . . . . . . . . . 10 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → (𝑤𝑤) ⊆ 𝑣)
7363, 72sstrd 3593 . . . . . . . . 9 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤𝑤𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
7473ex 450 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) ∧ 𝑢𝑈) ∧ (𝑢𝑢) ⊆ 𝑣) ∧ 𝑤𝑈) → ((𝑤 = 𝑤𝑤𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣))
7574reximdva 3011 . . . . . . 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 21924 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
78773adant2 1078 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑢𝑈 (𝑢𝑢) ⊆ 𝑣)
7976, 78r19.29a 3071 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃𝑋𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
8049, 50, 51, 79syl3anc 1323 . . . 4 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)
8148, 80r19.29a 3071 . . 3 (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) ∧ 𝑣𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
8281ralrimiva 2960 . 2 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ∀𝑣𝑈𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣)
83 iscfilu 22002 . . 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 956 1 ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  wss 3555  c0 3891  {csn 4148  cmpt 4673   × cxp 5072  ccnv 5073  ran crn 5075  cima 5077  ccom 5078  cfv 5847  Basecbs 15781  TopOpenctopn 16003  fBascfbas 19653  TopOnctopon 20618  TopSpctps 20619  neicnei 20811  Filcfil 21559  UnifOncust 21913  unifTopcutop 21944  UnifStcuss 21967  UnifSpcusp 21968  CauFiluccfilu 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-fbas 19662  df-top 20621  df-topon 20623  df-topsp 20624  df-nei 20812  df-fil 21560  df-ust 21914  df-utop 21945  df-usp 21971  df-cfilu 22001
This theorem is referenced by:  ucnextcn  22018
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