| Step | Hyp | Ref
| Expression |
| 1 | | simp2 1138 |
. . . . 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 22940 |
. . . . 5
⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋)) |
| 5 | 1, 4 | sylib 218 |
. . . 4
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
| 6 | | simp3 1139 |
. . . . 5
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋) |
| 7 | 6 | snssd 4809 |
. . . 4
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ⊆ 𝑋) |
| 8 | 6 | snn0d 4775 |
. . . 4
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → {𝑃} ≠ ∅) |
| 9 | | neifil 23888 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑃} ⊆ 𝑋 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋)) |
| 10 | 5, 7, 8, 9 | syl3anc 1373 |
. . 3
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋)) |
| 11 | | filfbas 23856 |
. . 3
⊢
(((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋)) |
| 12 | 10, 11 | syl 17 |
. 2
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋)) |
| 13 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃}) |
| 14 | | imaeq1 6073 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑤 → (𝑣 “ {𝑃}) = (𝑤 “ {𝑃})) |
| 15 | 14 | rspceeqv 3645 |
. . . . . . . . . 10
⊢ ((𝑤 ∈ 𝑈 ∧ (𝑤 “ {𝑃}) = (𝑤 “ {𝑃})) → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})) |
| 16 | 13, 15 | mpan2 691 |
. . . . . . . . 9
⊢ (𝑤 ∈ 𝑈 → ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})) |
| 17 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑤 ∈ V |
| 18 | 17 | imaex 7936 |
. . . . . . . . . 10
⊢ (𝑤 “ {𝑃}) ∈ V |
| 19 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) |
| 20 | 19 | elrnmpt 5969 |
. . . . . . . . . 10
⊢ ((𝑤 “ {𝑃}) ∈ V → ((𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃}))) |
| 21 | 18, 20 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})) ↔ ∃𝑣 ∈ 𝑈 (𝑤 “ {𝑃}) = (𝑣 “ {𝑃})) |
| 22 | 16, 21 | sylibr 234 |
. . . . . . . 8
⊢ (𝑤 ∈ 𝑈 → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 23 | 22 | ad2antlr 727 |
. . . . . . 7
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 24 | | neipcfilu.u |
. . . . . . . . . . . . 13
⊢ 𝑈 = (UnifSt‘𝑊) |
| 25 | 2, 24, 3 | isusp 24270 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 = (unifTop‘𝑈))) |
| 26 | 25 | simplbi 497 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ UnifSp → 𝑈 ∈ (UnifOn‘𝑋)) |
| 27 | 26 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 28 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(unifTop‘𝑈) =
(unifTop‘𝑈) |
| 29 | 28 | utopsnneip 24257 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 30 | 27, 6, 29 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘(unifTop‘𝑈))‘{𝑃}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃}))) |
| 31 | 30 | eleq2d 2827 |
. . . . . . . 8
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃}) ↔ (𝑤 “ {𝑃}) ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑃})))) |
| 32 | 31 | ad3antrrr 730 |
. . . . . . 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})))) |
| 33 | 23, 32 | mpbird 257 |
. . . . . 6
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘(unifTop‘𝑈))‘{𝑃})) |
| 34 | | simpl1 1192 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ (𝑣 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) → 𝑊 ∈ UnifSp) |
| 35 | 34 | 3anassrs 1361 |
. . . . . . . . 9
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝑊 ∈ UnifSp) |
| 36 | 25 | simprbi 496 |
. . . . . . . . 9
⊢ (𝑊 ∈ UnifSp → 𝐽 = (unifTop‘𝑈)) |
| 37 | 35, 36 | syl 17 |
. . . . . . . 8
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → 𝐽 = (unifTop‘𝑈)) |
| 38 | 37 | fveq2d 6910 |
. . . . . . 7
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (nei‘𝐽) = (nei‘(unifTop‘𝑈))) |
| 39 | 38 | fveq1d 6908 |
. . . . . 6
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((nei‘𝐽)‘{𝑃}) = ((nei‘(unifTop‘𝑈))‘{𝑃})) |
| 40 | 33, 39 | eleqtrrd 2844 |
. . . . 5
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → (𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃})) |
| 41 | | simpr 484 |
. . . . 5
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
| 42 | | id 22 |
. . . . . . . 8
⊢ (𝑎 = (𝑤 “ {𝑃}) → 𝑎 = (𝑤 “ {𝑃})) |
| 43 | 42 | sqxpeqd 5717 |
. . . . . . 7
⊢ (𝑎 = (𝑤 “ {𝑃}) → (𝑎 × 𝑎) = ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃}))) |
| 44 | 43 | sseq1d 4015 |
. . . . . 6
⊢ (𝑎 = (𝑤 “ {𝑃}) → ((𝑎 × 𝑎) ⊆ 𝑣 ↔ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) |
| 45 | 44 | rspcev 3622 |
. . . . 5
⊢ (((𝑤 “ {𝑃}) ∈ ((nei‘𝐽)‘{𝑃}) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
| 46 | 40, 41, 45 | syl2anc 584 |
. . . 4
⊢
(((((𝑊 ∈ UnifSp
∧ 𝑊 ∈ TopSp ∧
𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) ∧ ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
| 47 | 27 | adantr 480 |
. . . . 5
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 48 | 6 | adantr 480 |
. . . . 5
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑃 ∈ 𝑋) |
| 49 | | simpr 484 |
. . . . 5
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → 𝑣 ∈ 𝑈) |
| 50 | | simpll1 1213 |
. . . . . . . 8
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 51 | | simplr 769 |
. . . . . . . 8
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → 𝑢 ∈ 𝑈) |
| 52 | | ustexsym 24224 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) |
| 53 | 50, 51, 52 | syl2anc 584 |
. . . . . . 7
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) |
| 54 | 50 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑈 ∈ (UnifOn‘𝑋)) |
| 55 | | simplr 769 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ∈ 𝑈) |
| 56 | | ustssxp 24213 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋)) |
| 57 | 54, 55, 56 | syl2anc 584 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑤 ⊆ (𝑋 × 𝑋)) |
| 58 | | simpll2 1214 |
. . . . . . . . . . . 12
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ ((𝑢 ∘ 𝑢) ⊆ 𝑣 ∧ 𝑤 ∈ 𝑈 ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢))) → 𝑃 ∈ 𝑋) |
| 59 | 58 | 3anassrs 1361 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → 𝑃 ∈ 𝑋) |
| 60 | | ustneism 24232 |
. . . . . . . . . . 11
⊢ ((𝑤 ⊆ (𝑋 × 𝑋) ∧ 𝑃 ∈ 𝑋) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ (𝑤 ∘ ◡𝑤)) |
| 61 | 57, 59, 60 | syl2anc 584 |
. . . . . . . . . 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)) |
| 62 | | simprl 771 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ◡𝑤 = 𝑤) |
| 63 | 62 | coeq2d 5873 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ ◡𝑤) = (𝑤 ∘ 𝑤)) |
| 64 | | coss1 5866 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑤)) |
| 65 | | coss2 5867 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ⊆ 𝑢 → (𝑢 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢)) |
| 66 | 64, 65 | sstrd 3994 |
. . . . . . . . . . . . 13
⊢ (𝑤 ⊆ 𝑢 → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢)) |
| 67 | 66 | ad2antll 729 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ 𝑤) ⊆ (𝑢 ∘ 𝑢)) |
| 68 | | simpllr 776 |
. . . . . . . . . . . 12
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑢 ∘ 𝑢) ⊆ 𝑣) |
| 69 | 67, 68 | sstrd 3994 |
. . . . . . . . . . 11
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ 𝑤) ⊆ 𝑣) |
| 70 | 63, 69 | eqsstrd 4018 |
. . . . . . . . . 10
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → (𝑤 ∘ ◡𝑤) ⊆ 𝑣) |
| 71 | 61, 70 | sstrd 3994 |
. . . . . . . . 9
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢)) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
| 72 | 71 | ex 412 |
. . . . . . . 8
⊢
(((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) ∧ 𝑤 ∈ 𝑈) → ((◡𝑤 = 𝑤 ∧ 𝑤 ⊆ 𝑢) → ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣)) |
| 73 | 72 | reximdva 3168 |
. . . . . . 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)) |
| 74 | 53, 73 | mpd 15 |
. . . . . 6
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) ∧ 𝑢 ∈ 𝑈) ∧ (𝑢 ∘ 𝑢) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
| 75 | | ustexhalf 24219 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣) |
| 76 | 75 | 3adant2 1132 |
. . . . . 6
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑢 ∈ 𝑈 (𝑢 ∘ 𝑢) ⊆ 𝑣) |
| 77 | 74, 76 | r19.29a 3162 |
. . . . 5
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
| 78 | 47, 48, 49, 77 | syl3anc 1373 |
. . . 4
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 ((𝑤 “ {𝑃}) × (𝑤 “ {𝑃})) ⊆ 𝑣) |
| 79 | 46, 78 | r19.29a 3162 |
. . 3
⊢ (((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
| 80 | 79 | ralrimiva 3146 |
. 2
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣) |
| 81 | | iscfilu 24297 |
. . 3
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈) ↔ (((nei‘𝐽)‘{𝑃}) ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ ((nei‘𝐽)‘{𝑃})(𝑎 × 𝑎) ⊆ 𝑣))) |
| 82 | 27, 81 | 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))) |
| 83 | 12, 80, 82 | mpbir2and 713 |
1
⊢ ((𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝑃 ∈ 𝑋) → ((nei‘𝐽)‘{𝑃}) ∈ (CauFilu‘𝑈)) |