Step | Hyp | Ref
| Expression |
1 | | simp1 1137 |
. . . . 5
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝑈 ∈ (UnifOn‘𝑋)) |
2 | | simp2l 1200 |
. . . . 5
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (CauFilu‘𝑈)) |
3 | | iscfilu 23043 |
. . . . . 6
⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu‘𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣))) |
4 | 3 | biimpa 480 |
. . . . 5
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu‘𝑈)) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) |
5 | 1, 2, 4 | syl2anc 587 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣)) |
6 | 5 | simpld 498 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
7 | | simp3 1139 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → 𝐴 ⊆ 𝑋) |
8 | | simp2r 1201 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → ¬ ∅ ∈ (𝐹 ↾t 𝐴)) |
9 | | trfbas2 22597 |
. . . 4
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴))) |
10 | 9 | biimpar 481 |
. . 3
⊢ (((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ⊆ 𝑋) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴)) |
11 | 6, 7, 8, 10 | syl21anc 837 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (fBas‘𝐴)) |
12 | 2 | ad5antr 734 |
. . . . . . 7
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐹 ∈ (CauFilu‘𝑈)) |
13 | 1 | adantr 484 |
. . . . . . . . . 10
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑈 ∈ (UnifOn‘𝑋)) |
14 | 13 | elfvexd 6711 |
. . . . . . . . 9
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑋 ∈ V) |
15 | 7 | adantr 484 |
. . . . . . . . 9
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ⊆ 𝑋) |
16 | 14, 15 | ssexd 5193 |
. . . . . . . 8
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝐴 ∈ V) |
17 | 16 | ad4antr 732 |
. . . . . . 7
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝐴 ∈ V) |
18 | | simplr 769 |
. . . . . . 7
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑎 ∈ 𝐹) |
19 | | elrestr 16808 |
. . . . . . 7
⊢ ((𝐹 ∈
(CauFilu‘𝑈) ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐹) → (𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴)) |
20 | 12, 17, 18, 19 | syl3anc 1372 |
. . . . . 6
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴)) |
21 | | inxp 5676 |
. . . . . . 7
⊢ ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) |
22 | | simpr 488 |
. . . . . . . . 9
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → (𝑎 × 𝑎) ⊆ 𝑣) |
23 | 22 | ssrind 4127 |
. . . . . . . 8
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ (𝑣 ∩ (𝐴 × 𝐴))) |
24 | | simpllr 776 |
. . . . . . . 8
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) |
25 | 23, 24 | sseqtrrd 3919 |
. . . . . . 7
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 × 𝑎) ∩ (𝐴 × 𝐴)) ⊆ 𝑤) |
26 | 21, 25 | eqsstrrid 3927 |
. . . . . 6
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤) |
27 | | id 22 |
. . . . . . . . 9
⊢ (𝑏 = (𝑎 ∩ 𝐴) → 𝑏 = (𝑎 ∩ 𝐴)) |
28 | 27 | sqxpeqd 5558 |
. . . . . . . 8
⊢ (𝑏 = (𝑎 ∩ 𝐴) → (𝑏 × 𝑏) = ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴))) |
29 | 28 | sseq1d 3909 |
. . . . . . 7
⊢ (𝑏 = (𝑎 ∩ 𝐴) → ((𝑏 × 𝑏) ⊆ 𝑤 ↔ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤)) |
30 | 29 | rspcev 3527 |
. . . . . 6
⊢ (((𝑎 ∩ 𝐴) ∈ (𝐹 ↾t 𝐴) ∧ ((𝑎 ∩ 𝐴) × (𝑎 ∩ 𝐴)) ⊆ 𝑤) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤) |
31 | 20, 26, 30 | syl2anc 587 |
. . . . 5
⊢
(((((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) ∧ 𝑎 ∈ 𝐹) ∧ (𝑎 × 𝑎) ⊆ 𝑣) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤) |
32 | 5 | simprd 499 |
. . . . . . 7
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → ∀𝑣 ∈ 𝑈 ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣) |
33 | 32 | r19.21bi 3122 |
. . . . . 6
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣) |
34 | 33 | ad4ant13 751 |
. . . . 5
⊢
(((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑎 ∈ 𝐹 (𝑎 × 𝑎) ⊆ 𝑣) |
35 | 31, 34 | r19.29a 3200 |
. . . 4
⊢
(((((𝑈 ∈
(UnifOn‘𝑋) ∧
(𝐹 ∈
(CauFilu‘𝑈) ∧ ¬ ∅ ∈ (𝐹 ↾t 𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤) |
36 | 16, 16 | xpexd 7495 |
. . . . 5
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → (𝐴 × 𝐴) ∈ V) |
37 | | simpr 488 |
. . . . 5
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) |
38 | | elrest 16807 |
. . . . . 6
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) → (𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴)) ↔ ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴)))) |
39 | 38 | biimpa 480 |
. . . . 5
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐴 × 𝐴) ∈ V) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) |
40 | 13, 36, 37, 39 | syl21anc 837 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑣 ∈ 𝑈 𝑤 = (𝑣 ∩ (𝐴 × 𝐴))) |
41 | 35, 40 | r19.29a 3200 |
. . 3
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) ∧ 𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))) → ∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤) |
42 | 41 | ralrimiva 3097 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤) |
43 | | trust 22984 |
. . . 4
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴)) |
44 | 1, 7, 43 | syl2anc 587 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴)) |
45 | | iscfilu 23043 |
. . 3
⊢ ((𝑈 ↾t (𝐴 × 𝐴)) ∈ (UnifOn‘𝐴) → ((𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤))) |
46 | 44, 45 | syl 17 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴))) ↔ ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ∧ ∀𝑤 ∈ (𝑈 ↾t (𝐴 × 𝐴))∃𝑏 ∈ (𝐹 ↾t 𝐴)(𝑏 × 𝑏) ⊆ 𝑤))) |
47 | 11, 42, 46 | mpbir2and 713 |
1
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝐹 ∈ (CauFilu‘𝑈) ∧ ¬ ∅ ∈
(𝐹 ↾t
𝐴)) ∧ 𝐴 ⊆ 𝑋) → (𝐹 ↾t 𝐴) ∈ (CauFilu‘(𝑈 ↾t (𝐴 × 𝐴)))) |