Step | Hyp | Ref
| Expression |
1 | | simp2 1055 |
. . . . 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 20551 |
. . . . 5
⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 1, 4 | sylib 207 |
. . . 4
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | | simp3 1056 |
. . . . 5
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋) |
7 | 6 | snssd 4281 |
. . . 4
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ⊆ 𝑋) |
8 | | snnzg 4251 |
. . . . 5
⊢ (𝑃 ∈ 𝑋 → {𝑃} ≠ ∅) |
9 | 6, 8 | syl 17 |
. . . 4
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ≠ ∅) |
10 | | neifil 21494 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋)) |
11 | 5, 7, 9, 10 | syl3anc 1318 |
. . 3
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋)) |
12 | | filfbas 21462 |
. . 3
⊢
(((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋)) |
13 | 11, 12 | syl 17 |
. 2
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋)) |
14 | | eqid 2610 |
. . . . . . . . . 10
⊢ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃}) |
15 | | imaeq1 5380 |
. . . . . . . . . . . 12
⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃})) |
16 | 15 | eqeq2d 2620 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑤 → ((𝑤 “ {𝑃}) = (𝑣 “ {𝑃}) ↔ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃}))) |
17 | 16 | rspcev 3282 |
. . . . . . . . . 10
⊢ ((𝑤 ∈ 𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})) |
18 | 14, 17 | mpan2 703 |
. . . . . . . . 9
⊢ (𝑤 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})) |
19 | | vex 3176 |
. . . . . . . . . . 11
⊢ 𝑤 ∈ V |
20 | 19 | imaex 6996 |
. . . . . . . . . 10
⊢ (𝑤 “ {𝑃}) ∈ V |
21 | | eqid 2610 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) |
22 | 21 | elrnmpt 5293 |
. . . . . . . . . 10
⊢ ((𝑤 “ {𝑃}) ∈ V → ((𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))) |
23 | 20, 22 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})) |
24 | 18, 23 | sylibr 223 |
. . . . . . . 8
⊢ (𝑤 ∈ 𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
25 | 24 | ad2antlr 759 |
. . . . . . 7
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
26 | | neipcfilu.u |
. . . . . . . . . . . . 13
⊢ 𝑈 = (UnifSt‘𝑊) |
27 | 2, 26, 3 | isusp 21875 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈))) |
28 | 27 | simplbi 475 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋)) |
29 | 28 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋)) |
30 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(unifTop‘𝑈) =
(unifTop‘𝑈) |
31 | 30 | utopsnneip 21862 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
32 | 29, 6, 31 | syl2anc 691 |
. . . . . . . . 9
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
33 | 32 | eleq2d 2673 |
. . . . . . . 8
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))) |
34 | 33 | ad3antrrr 762 |
. . . . . . 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})))) |
35 | 25, 34 | mpbird 246 |
. . . . . 6
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃})) |
36 | | simpl1 1057 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ (𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp) |
37 | 36 | 3anassrs 1282 |
. . . . . . . . 9
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp) |
38 | 27 | simprbi 479 |
. . . . . . . . 9
⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘𝑈)) |
39 | 37, 38 | syl 17 |
. . . . . . . 8
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝐽 = (unifTop‘𝑈)) |
40 | 39 | fveq2d 6107 |
. . . . . . 7
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈))) |
41 | 40 | fveq1d 6105 |
. . . . . 6
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃})) |
42 | 35, 41 | eleqtrrd 2691 |
. . . . 5
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃})) |
43 | | simpr 476 |
. . . . 5
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
44 | | id 22 |
. . . . . . . 8
⊢ (𝑎 = (𝑤 “ {𝑃}) → 𝑎 = (𝑤 “ {𝑃})) |
45 | 44 | sqxpeqd 5065 |
. . . . . . 7
⊢ (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃}))) |
46 | 45 | sseq1d 3595 |
. . . . . 6
⊢ (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) |
47 | 46 | rspcev 3282 |
. . . . 5
⊢ (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
48 | 42, 43, 47 | syl2anc 691 |
. . . 4
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
49 | 29 | adantr 480 |
. . . . 5
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
50 | 6 | adantr 480 |
. . . . 5
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑃 ∈ 𝑋) |
51 | | simpr 476 |
. . . . 5
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑈) |
52 | | simpll1 1093 |
. . . . . . . 8
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋)) |
53 | | simplr 788 |
. . . . . . . 8
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝑈) |
54 | | ustexsym 21829 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) |
55 | 52, 53, 54 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) |
56 | 52 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑈 ∈ (UnifOn‘𝑋)) |
57 | | simplr 788 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ∈ 𝑈) |
58 | | ustssxp 21818 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋)) |
59 | 56, 57, 58 | syl2anc 691 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋)) |
60 | | simpll2 1094 |
. . . . . . . . . . . 12
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ ((𝑢 ∘ 𝑢) ⊆ 𝑣 ∧ 𝑤 ∈ 𝑈 ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))) → 𝑃 ∈ 𝑋) |
61 | 60 | 3anassrs 1282 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑃 ∈ 𝑋) |
62 | | ustneism 21837 |
. . . . . . . . . . 11
⊢ ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤 ∘ ◡𝑤)) |
63 | 59, 61, 62 | syl2anc 691 |
. . . . . . . . . 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)) |
64 | | simprl 790 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ◡𝑤 = 𝑤) |
65 | 64 | coeq2d 5206 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ ◡𝑤) = (𝑤 ∘ 𝑤)) |
66 | | coss1 5199 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑤)) |
67 | | coss2 5200 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ⊆ 𝑢 → (𝑢 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢)) |
68 | 66, 67 | sstrd 3578 |
. . . . . . . . . . . . 13
⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢)) |
69 | 68 | ad2antll 761 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢)) |
70 | | simpllr 795 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑢 ∘ 𝑢) ⊆ 𝑣) |
71 | 69, 70 | sstrd 3578 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ 𝑤) ⊆ 𝑣) |
72 | 65, 71 | eqsstrd 3602 |
. . . . . . . . . 10
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ ◡𝑤) ⊆ 𝑣) |
73 | 63, 72 | sstrd 3578 |
. . . . . . . . 9
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
74 | 73 | ex 449 |
. . . . . . . 8
⊢
(((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) |
75 | 74 | reximdva 3000 |
. . . . . . 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)) |
76 | 55, 75 | mpd 15 |
. . . . . 6
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
77 | | ustexhalf 21824 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣) |
78 | 77 | 3adant2 1073 |
. . . . . 6
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣) |
79 | 76, 78 | r19.29a 3060 |
. . . . 5
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
80 | 49, 50, 51, 79 | syl3anc 1318 |
. . . 4
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
81 | 48, 80 | r19.29a 3060 |
. . 3
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
82 | 81 | ralrimiva 2949 |
. 2
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
83 | | iscfilu 21902 |
. . 3
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣))) |
84 | 29, 83 | syl 17 |
. 2
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣))) |
85 | 13, 82, 84 | mpbir2and 959 |
1
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈)) |