Step | Hyp | Ref
| Expression |
1 | | xkococn.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ (𝑅 Cn 𝑆)) |
2 | | xkococn.c |
. . . 4
⊢ (𝜑 → (𝑅 ↾t 𝐾) ∈ Comp) |
3 | | imacmp 21200 |
. . . 4
⊢ ((𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝑅 ↾t 𝐾) ∈ Comp) → (𝑆 ↾t (𝐵 “ 𝐾)) ∈ Comp) |
4 | 1, 2, 3 | syl2anc 693 |
. . 3
⊢ (𝜑 → (𝑆 ↾t (𝐵 “ 𝐾)) ∈ Comp) |
5 | | xkococn.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ 𝑛-Locally
Comp) |
6 | 5 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → 𝑆 ∈ 𝑛-Locally
Comp) |
7 | | xkococn.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ (𝑆 Cn 𝑇)) |
8 | | xkococn.v |
. . . . . . . . . 10
⊢ (𝜑 → 𝑉 ∈ 𝑇) |
9 | | cnima 21069 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ (𝑆 Cn 𝑇) ∧ 𝑉 ∈ 𝑇) → (◡𝐴 “ 𝑉) ∈ 𝑆) |
10 | 7, 8, 9 | syl2anc 693 |
. . . . . . . . 9
⊢ (𝜑 → (◡𝐴 “ 𝑉) ∈ 𝑆) |
11 | 10 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → (◡𝐴 “ 𝑉) ∈ 𝑆) |
12 | | imaco 5640 |
. . . . . . . . . . 11
⊢ ((𝐴 ∘ 𝐵) “ 𝐾) = (𝐴 “ (𝐵 “ 𝐾)) |
13 | | xkococn.i |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐴 ∘ 𝐵) “ 𝐾) ⊆ 𝑉) |
14 | 12, 13 | syl5eqssr 3650 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 “ (𝐵 “ 𝐾)) ⊆ 𝑉) |
15 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑆 =
∪ 𝑆 |
16 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑇 =
∪ 𝑇 |
17 | 15, 16 | cnf 21050 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ (𝑆 Cn 𝑇) → 𝐴:∪ 𝑆⟶∪ 𝑇) |
18 | | ffun 6048 |
. . . . . . . . . . . 12
⊢ (𝐴:∪
𝑆⟶∪ 𝑇
→ Fun 𝐴) |
19 | 7, 17, 18 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → Fun 𝐴) |
20 | | imassrn 5477 |
. . . . . . . . . . . . 13
⊢ (𝐵 “ 𝐾) ⊆ ran 𝐵 |
21 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑅 =
∪ 𝑅 |
22 | 21, 15 | cnf 21050 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈ (𝑅 Cn 𝑆) → 𝐵:∪ 𝑅⟶∪ 𝑆) |
23 | | frn 6053 |
. . . . . . . . . . . . . 14
⊢ (𝐵:∪
𝑅⟶∪ 𝑆
→ ran 𝐵 ⊆ ∪ 𝑆) |
24 | 1, 22, 23 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ran 𝐵 ⊆ ∪ 𝑆) |
25 | 20, 24 | syl5ss 3614 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵 “ 𝐾) ⊆ ∪ 𝑆) |
26 | | fdm 6051 |
. . . . . . . . . . . . 13
⊢ (𝐴:∪
𝑆⟶∪ 𝑇
→ dom 𝐴 = ∪ 𝑆) |
27 | 7, 17, 26 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → dom 𝐴 = ∪ 𝑆) |
28 | 25, 27 | sseqtr4d 3642 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵 “ 𝐾) ⊆ dom 𝐴) |
29 | | funimass3 6333 |
. . . . . . . . . . 11
⊢ ((Fun
𝐴 ∧ (𝐵 “ 𝐾) ⊆ dom 𝐴) → ((𝐴 “ (𝐵 “ 𝐾)) ⊆ 𝑉 ↔ (𝐵 “ 𝐾) ⊆ (◡𝐴 “ 𝑉))) |
30 | 19, 28, 29 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 “ (𝐵 “ 𝐾)) ⊆ 𝑉 ↔ (𝐵 “ 𝐾) ⊆ (◡𝐴 “ 𝑉))) |
31 | 14, 30 | mpbid 222 |
. . . . . . . . 9
⊢ (𝜑 → (𝐵 “ 𝐾) ⊆ (◡𝐴 “ 𝑉)) |
32 | 31 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → 𝑥 ∈ (◡𝐴 “ 𝑉)) |
33 | | nlly2i 21279 |
. . . . . . . 8
⊢ ((𝑆 ∈ 𝑛-Locally Comp
∧ (◡𝐴 “ 𝑉) ∈ 𝑆 ∧ 𝑥 ∈ (◡𝐴 “ 𝑉)) → ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp)) |
34 | 6, 11, 32, 33 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp)) |
35 | | nllytop 21276 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ 𝑛-Locally Comp
→ 𝑆 ∈
Top) |
36 | 5, 35 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ Top) |
37 | 36 | ad3antrrr 766 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑆 ∈ Top) |
38 | | imaexg 7103 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∈ (𝑅 Cn 𝑆) → (𝐵 “ 𝐾) ∈ V) |
39 | 1, 38 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐵 “ 𝐾) ∈ V) |
40 | 39 | ad3antrrr 766 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → (𝐵 “ 𝐾) ∈ V) |
41 | | simprl 794 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑢 ∈ 𝑆) |
42 | | elrestr 16089 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Top ∧ (𝐵 “ 𝐾) ∈ V ∧ 𝑢 ∈ 𝑆) → (𝑢 ∩ (𝐵 “ 𝐾)) ∈ (𝑆 ↾t (𝐵 “ 𝐾))) |
43 | 37, 40, 41, 42 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵 “ 𝐾)) ∈ (𝑆 ↾t (𝐵 “ 𝐾))) |
44 | | simprr1 1109 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑥 ∈ 𝑢) |
45 | | simpllr 799 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝐵 “ 𝐾)) |
46 | 44, 45 | elind 3798 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾))) |
47 | | inss1 3833 |
. . . . . . . . . . . 12
⊢ (𝑢 ∩ (𝐵 “ 𝐾)) ⊆ 𝑢 |
48 | | elpwi 4168 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉) → 𝑠 ⊆ (◡𝐴 “ 𝑉)) |
49 | 48 | ad2antlr 763 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑠 ⊆ (◡𝐴 “ 𝑉)) |
50 | | elssuni 4467 |
. . . . . . . . . . . . . . . 16
⊢ ((◡𝐴 “ 𝑉) ∈ 𝑆 → (◡𝐴 “ 𝑉) ⊆ ∪ 𝑆) |
51 | 10, 50 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (◡𝐴 “ 𝑉) ⊆ ∪ 𝑆) |
52 | 51 | ad3antrrr 766 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → (◡𝐴 “ 𝑉) ⊆ ∪ 𝑆) |
53 | 49, 52 | sstrd 3613 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑠 ⊆ ∪ 𝑆) |
54 | | simprr2 1110 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑢 ⊆ 𝑠) |
55 | 15 | ssntr 20862 |
. . . . . . . . . . . . 13
⊢ (((𝑆 ∈ Top ∧ 𝑠 ⊆ ∪ 𝑆)
∧ (𝑢 ∈ 𝑆 ∧ 𝑢 ⊆ 𝑠)) → 𝑢 ⊆ ((int‘𝑆)‘𝑠)) |
56 | 37, 53, 41, 54, 55 | syl22anc 1327 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → 𝑢 ⊆ ((int‘𝑆)‘𝑠)) |
57 | 47, 56 | syl5ss 3614 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → (𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠)) |
58 | | simprr3 1111 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → (𝑆 ↾t 𝑠) ∈ Comp) |
59 | 57, 58 | jca 554 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) |
60 | | eleq2 2690 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾)))) |
61 | | sseq1 3626 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ (𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠))) |
62 | 61 | anbi1d 741 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp) ↔ ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
63 | 60, 62 | anbi12d 747 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝑢 ∩ (𝐵 “ 𝐾)) → ((𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾)) ∧ ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)))) |
64 | 63 | rspcev 3309 |
. . . . . . . . . 10
⊢ (((𝑢 ∩ (𝐵 “ 𝐾)) ∈ (𝑆 ↾t (𝐵 “ 𝐾)) ∧ (𝑥 ∈ (𝑢 ∩ (𝐵 “ 𝐾)) ∧ ((𝑢 ∩ (𝐵 “ 𝐾)) ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
65 | 43, 46, 59, 64 | syl12anc 1324 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) ∧ (𝑢 ∈ 𝑆 ∧ (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
66 | 65 | rexlimdvaa 3032 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) ∧ 𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)) → (∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp) → ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)))) |
67 | 66 | reximdva 3017 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → (∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)∃𝑢 ∈ 𝑆 (𝑥 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑠 ∧ (𝑆 ↾t 𝑠) ∈ Comp) → ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)))) |
68 | 34, 67 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
69 | | rexcom 3099 |
. . . . . . 7
⊢
(∃𝑠 ∈
𝒫 (◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
70 | | r19.42v 3092 |
. . . . . . . 8
⊢
(∃𝑠 ∈
𝒫 (◡𝐴 “ 𝑉)(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) ↔ (𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
71 | 70 | rexbii 3041 |
. . . . . . 7
⊢
(∃𝑦 ∈
(𝑆 ↾t
(𝐵 “ 𝐾))∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
72 | 69, 71 | bitri 264 |
. . . . . 6
⊢
(∃𝑠 ∈
𝒫 (◡𝐴 “ 𝑉)∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ (𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) ↔ ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
73 | 68, 72 | sylib 208 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 “ 𝐾)) → ∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
74 | 73 | ralrimiva 2966 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ (𝐵 “ 𝐾)∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
75 | 15 | restuni 20966 |
. . . . . 6
⊢ ((𝑆 ∈ Top ∧ (𝐵 “ 𝐾) ⊆ ∪ 𝑆) → (𝐵 “ 𝐾) = ∪ (𝑆 ↾t (𝐵 “ 𝐾))) |
76 | 36, 25, 75 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (𝐵 “ 𝐾) = ∪ (𝑆 ↾t (𝐵 “ 𝐾))) |
77 | 76 | raleqdv 3144 |
. . . 4
⊢ (𝜑 → (∀𝑥 ∈ (𝐵 “ 𝐾)∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)) ↔ ∀𝑥 ∈ ∪ (𝑆
↾t (𝐵
“ 𝐾))∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp)))) |
78 | 74, 77 | mpbid 222 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ∪ (𝑆 ↾t (𝐵 “ 𝐾))∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) |
79 | | eqid 2622 |
. . . 4
⊢ ∪ (𝑆
↾t (𝐵
“ 𝐾)) = ∪ (𝑆
↾t (𝐵
“ 𝐾)) |
80 | | fveq2 6191 |
. . . . . 6
⊢ (𝑠 = (𝑘‘𝑦) → ((int‘𝑆)‘𝑠) = ((int‘𝑆)‘(𝑘‘𝑦))) |
81 | 80 | sseq2d 3633 |
. . . . 5
⊢ (𝑠 = (𝑘‘𝑦) → (𝑦 ⊆ ((int‘𝑆)‘𝑠) ↔ 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)))) |
82 | | oveq2 6658 |
. . . . . 6
⊢ (𝑠 = (𝑘‘𝑦) → (𝑆 ↾t 𝑠) = (𝑆 ↾t (𝑘‘𝑦))) |
83 | 82 | eleq1d 2686 |
. . . . 5
⊢ (𝑠 = (𝑘‘𝑦) → ((𝑆 ↾t 𝑠) ∈ Comp ↔ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)) |
84 | 81, 83 | anbi12d 747 |
. . . 4
⊢ (𝑠 = (𝑘‘𝑦) → ((𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp) ↔ (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp))) |
85 | 79, 84 | cmpcovf 21194 |
. . 3
⊢ (((𝑆 ↾t (𝐵 “ 𝐾)) ∈ Comp ∧ ∀𝑥 ∈ ∪ (𝑆
↾t (𝐵
“ 𝐾))∃𝑦 ∈ (𝑆 ↾t (𝐵 “ 𝐾))(𝑥 ∈ 𝑦 ∧ ∃𝑠 ∈ 𝒫 (◡𝐴 “ 𝑉)(𝑦 ⊆ ((int‘𝑆)‘𝑠) ∧ (𝑆 ↾t 𝑠) ∈ Comp))) → ∃𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)(∪
(𝑆 ↾t
(𝐵 “ 𝐾)) = ∪ 𝑤
∧ ∃𝑘(𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) |
86 | 4, 78, 85 | syl2anc 693 |
. 2
⊢ (𝜑 → ∃𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)(∪
(𝑆 ↾t
(𝐵 “ 𝐾)) = ∪ 𝑤
∧ ∃𝑘(𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) |
87 | 76 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) → (𝐵 “ 𝐾) = ∪ (𝑆 ↾t (𝐵 “ 𝐾))) |
88 | 87 | eqeq1d 2624 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) → ((𝐵 “ 𝐾) = ∪ 𝑤 ↔ ∪ (𝑆
↾t (𝐵
“ 𝐾)) = ∪ 𝑤)) |
89 | 88 | biimpar 502 |
. . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ∪ (𝑆
↾t (𝐵
“ 𝐾)) = ∪ 𝑤)
→ (𝐵 “ 𝐾) = ∪
𝑤) |
90 | 36 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑆 ∈ Top) |
91 | | cntop2 21045 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ (𝑆 Cn 𝑇) → 𝑇 ∈ Top) |
92 | 7, 91 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ∈ Top) |
93 | 92 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑇 ∈ Top) |
94 | | xkotop 21391 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ Top ∧ 𝑇 ∈ Top) → (𝑇 ^ko 𝑆) ∈ Top) |
95 | 90, 93, 94 | syl2anc 693 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝑇 ^ko 𝑆) ∈ Top) |
96 | | cntop1 21044 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ (𝑅 Cn 𝑆) → 𝑅 ∈ Top) |
97 | 1, 96 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ Top) |
98 | 97 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑅 ∈ Top) |
99 | | xkotop 21391 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top) |
100 | 98, 90, 99 | syl2anc 693 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ^ko 𝑅) ∈ Top) |
101 | | simprrl 804 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉)) |
102 | | frn 6053 |
. . . . . . . . . . . . 13
⊢ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) → ran 𝑘 ⊆ 𝒫 (◡𝐴 “ 𝑉)) |
103 | 101, 102 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ran 𝑘 ⊆ 𝒫 (◡𝐴 “ 𝑉)) |
104 | | sspwuni 4611 |
. . . . . . . . . . . 12
⊢ (ran
𝑘 ⊆ 𝒫 (◡𝐴 “ 𝑉) ↔ ∪ ran
𝑘 ⊆ (◡𝐴 “ 𝑉)) |
105 | 103, 104 | sylib 208 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ ran 𝑘 ⊆ (◡𝐴 “ 𝑉)) |
106 | 10 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (◡𝐴 “ 𝑉) ∈ 𝑆) |
107 | 106, 50 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (◡𝐴 “ 𝑉) ⊆ ∪ 𝑆) |
108 | 105, 107 | sstrd 3613 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ ran 𝑘 ⊆ ∪ 𝑆) |
109 | | ffn 6045 |
. . . . . . . . . . . . 13
⊢ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) → 𝑘 Fn 𝑤) |
110 | | fniunfv 6505 |
. . . . . . . . . . . . 13
⊢ (𝑘 Fn 𝑤 → ∪
𝑦 ∈ 𝑤 (𝑘‘𝑦) = ∪ ran 𝑘) |
111 | 101, 109,
110 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) = ∪ ran 𝑘) |
112 | 111 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↾t ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) = (𝑆 ↾t ∪ ran 𝑘)) |
113 | | inss2 3834 |
. . . . . . . . . . . . 13
⊢
(𝒫 (𝑆
↾t (𝐵
“ 𝐾)) ∩ Fin)
⊆ Fin |
114 | | simplr 792 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) |
115 | 113, 114 | sseldi 3601 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑤 ∈ Fin) |
116 | | simprrr 805 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)) |
117 | | simpr 477 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) → (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) |
118 | 117 | ralimi 2952 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 ∈
𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) → ∀𝑦 ∈ 𝑤 (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) |
119 | 116, 118 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) |
120 | 15 | fiuncmp 21207 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ Top ∧ 𝑤 ∈ Fin ∧ ∀𝑦 ∈ 𝑤 (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) → (𝑆 ↾t ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) ∈ Comp) |
121 | 90, 115, 119, 120 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↾t ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) ∈ Comp) |
122 | 112, 121 | eqeltrrd 2702 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝑆 ↾t ∪ ran 𝑘) ∈ Comp) |
123 | 8 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑉 ∈ 𝑇) |
124 | 15, 90, 93, 108, 122, 123 | xkoopn 21392 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ∈ (𝑇 ^ko 𝑆)) |
125 | | xkococn.k |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ⊆ ∪ 𝑅) |
126 | 125 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝐾 ⊆ ∪ 𝑅) |
127 | 2 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝑅 ↾t 𝐾) ∈ Comp) |
128 | 111, 108 | eqsstrd 3639 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆) |
129 | | iunss 4561 |
. . . . . . . . . . . . 13
⊢ (∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆 ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆) |
130 | 128, 129 | sylib 208 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆) |
131 | 15 | ntropn 20853 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ Top ∧ (𝑘‘𝑦) ⊆ ∪ 𝑆) → ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆) |
132 | 131 | ex 450 |
. . . . . . . . . . . . 13
⊢ (𝑆 ∈ Top → ((𝑘‘𝑦) ⊆ ∪ 𝑆 → ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆)) |
133 | 132 | ralimdv 2963 |
. . . . . . . . . . . 12
⊢ (𝑆 ∈ Top →
(∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆 → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆)) |
134 | 90, 130, 133 | sylc 65 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆) |
135 | | iunopn 20703 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ Top ∧ ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆) → ∪
𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆) |
136 | 90, 134, 135 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ∈ 𝑆) |
137 | 21, 98, 90, 126, 127, 136 | xkoopn 21392 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ∈ (𝑆 ^ko 𝑅)) |
138 | | txopn 21405 |
. . . . . . . . 9
⊢ ((((𝑇 ^ko 𝑆) ∈ Top ∧ (𝑆 ^ko 𝑅) ∈ Top) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ∈ (𝑇 ^ko 𝑆) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ∈ (𝑆 ^ko 𝑅))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))) |
139 | 95, 100, 124, 137, 138 | syl22anc 1327 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))) |
140 | 7 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝐴 ∈ (𝑆 Cn 𝑇)) |
141 | | imaiun 6503 |
. . . . . . . . . . . 12
⊢ (𝐴 “ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) = ∪
𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) |
142 | 111 | imaeq2d 5466 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝐴 “ ∪
𝑦 ∈ 𝑤 (𝑘‘𝑦)) = (𝐴 “ ∪ ran
𝑘)) |
143 | 141, 142 | syl5eqr 2670 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) = (𝐴 “ ∪ ran
𝑘)) |
144 | 111, 105 | eqsstrd 3639 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉)) |
145 | 19 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → Fun 𝐴) |
146 | 101, 109 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝑘 Fn 𝑤) |
147 | 27 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → dom 𝐴 = ∪
𝑆) |
148 | 108, 147 | sseqtr4d 3642 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ ran 𝑘 ⊆ dom 𝐴) |
149 | | simpl1 1064 |
. . . . . . . . . . . . . . . 16
⊢ (((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦 ∈ 𝑤) → Fun 𝐴) |
150 | 110 | 3ad2ant2 1083 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∪
𝑦 ∈ 𝑤 (𝑘‘𝑦) = ∪ ran 𝑘) |
151 | | simp3 1063 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∪ ran
𝑘 ⊆ dom 𝐴) |
152 | 150, 151 | eqsstrd 3639 |
. . . . . . . . . . . . . . . . . 18
⊢ ((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∪
𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴) |
153 | | iunss 4561 |
. . . . . . . . . . . . . . . . . 18
⊢ (∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴 ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴) |
154 | 152, 153 | sylib 208 |
. . . . . . . . . . . . . . . . 17
⊢ ((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ dom 𝐴) |
155 | 154 | r19.21bi 2932 |
. . . . . . . . . . . . . . . 16
⊢ (((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦 ∈ 𝑤) → (𝑘‘𝑦) ⊆ dom 𝐴) |
156 | | funimass3 6333 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun
𝐴 ∧ (𝑘‘𝑦) ⊆ dom 𝐴) → ((𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉))) |
157 | 149, 155,
156 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) ∧ 𝑦 ∈ 𝑤) → ((𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉))) |
158 | 157 | ralbidva 2985 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → (∀𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉))) |
159 | | iunss 4561 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∀𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉) |
160 | | iunss 4561 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉) ↔ ∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉)) |
161 | 158, 159,
160 | 3bitr4g 303 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐴 ∧ 𝑘 Fn 𝑤 ∧ ∪ ran 𝑘 ⊆ dom 𝐴) → (∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∪
𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉))) |
162 | 145, 146,
148, 161 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉 ↔ ∪
𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ (◡𝐴 “ 𝑉))) |
163 | 144, 162 | mpbird 247 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 (𝐴 “ (𝑘‘𝑦)) ⊆ 𝑉) |
164 | 143, 163 | eqsstr3d 3640 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝐴 “ ∪ ran
𝑘) ⊆ 𝑉) |
165 | | imaeq1 5461 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐴 → (𝑎 “ ∪ ran
𝑘) = (𝐴 “ ∪ ran
𝑘)) |
166 | 165 | sseq1d 3632 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → ((𝑎 “ ∪ ran
𝑘) ⊆ 𝑉 ↔ (𝐴 “ ∪ ran
𝑘) ⊆ 𝑉)) |
167 | 166 | elrab 3363 |
. . . . . . . . . 10
⊢ (𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ↔ (𝐴 ∈ (𝑆 Cn 𝑇) ∧ (𝐴 “ ∪ ran
𝑘) ⊆ 𝑉)) |
168 | 140, 164,
167 | sylanbrc 698 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉}) |
169 | 1 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝐵 ∈ (𝑅 Cn 𝑆)) |
170 | | simprl 794 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝐵 “ 𝐾) = ∪ 𝑤) |
171 | | uniiun 4573 |
. . . . . . . . . . . 12
⊢ ∪ 𝑤 =
∪ 𝑦 ∈ 𝑤 𝑦 |
172 | 170, 171 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝐵 “ 𝐾) = ∪
𝑦 ∈ 𝑤 𝑦) |
173 | | simpl 473 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) → 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦))) |
174 | 173 | ralimi 2952 |
. . . . . . . . . . . 12
⊢
(∀𝑦 ∈
𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp) → ∀𝑦 ∈ 𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦))) |
175 | | ss2iun 4536 |
. . . . . . . . . . . 12
⊢
(∀𝑦 ∈
𝑤 𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) → ∪
𝑦 ∈ 𝑤 𝑦 ⊆ ∪
𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))) |
176 | 116, 174,
175 | 3syl 18 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 𝑦 ⊆ ∪
𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))) |
177 | 172, 176 | eqsstrd 3639 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝐵 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))) |
178 | | imaeq1 5461 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝐵 → (𝑏 “ 𝐾) = (𝐵 “ 𝐾)) |
179 | 178 | sseq1d 3632 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐵 → ((𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ↔ (𝐵 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))) |
180 | 179 | elrab 3363 |
. . . . . . . . . 10
⊢ (𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ↔ (𝐵 ∈ (𝑅 Cn 𝑆) ∧ (𝐵 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))) |
181 | 169, 177,
180 | sylanbrc 698 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) |
182 | | opelxpi 5148 |
. . . . . . . . 9
⊢ ((𝐴 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ∧ 𝐵 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → 〈𝐴, 𝐵〉 ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) |
183 | 168, 181,
182 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → 〈𝐴, 𝐵〉 ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) |
184 | | imaeq1 5461 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑢 → (𝑎 “ ∪ ran
𝑘) = (𝑢 “ ∪ ran
𝑘)) |
185 | 184 | sseq1d 3632 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑢 → ((𝑎 “ ∪ ran
𝑘) ⊆ 𝑉 ↔ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉)) |
186 | 185 | elrab 3363 |
. . . . . . . . . . . . 13
⊢ (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ↔ (𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉)) |
187 | | imaeq1 5461 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑣 → (𝑏 “ 𝐾) = (𝑣 “ 𝐾)) |
188 | 187 | sseq1d 3632 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑣 → ((𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ↔ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))) |
189 | 188 | elrab 3363 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ↔ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)))) |
190 | 186, 189 | anbi12i 733 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ∧ 𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ↔ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) |
191 | | simprll 802 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → 𝑢 ∈ (𝑆 Cn 𝑇)) |
192 | | simprrl 804 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → 𝑣 ∈ (𝑅 Cn 𝑆)) |
193 | | coeq1 5279 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑢 → (𝑓 ∘ 𝑔) = (𝑢 ∘ 𝑔)) |
194 | | coeq2 5280 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑣 → (𝑢 ∘ 𝑔) = (𝑢 ∘ 𝑣)) |
195 | | xkococn.1 |
. . . . . . . . . . . . . . 15
⊢ 𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓 ∘ 𝑔)) |
196 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑢 ∈ V |
197 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑣 ∈ V |
198 | 196, 197 | coex 7118 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 ∘ 𝑣) ∈ V |
199 | 193, 194,
195, 198 | ovmpt2 6796 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ 𝑣 ∈ (𝑅 Cn 𝑆)) → (𝑢𝐹𝑣) = (𝑢 ∘ 𝑣)) |
200 | 191, 192,
199 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢𝐹𝑣) = (𝑢 ∘ 𝑣)) |
201 | | cnco 21070 |
. . . . . . . . . . . . . . 15
⊢ ((𝑣 ∈ (𝑅 Cn 𝑆) ∧ 𝑢 ∈ (𝑆 Cn 𝑇)) → (𝑢 ∘ 𝑣) ∈ (𝑅 Cn 𝑇)) |
202 | 192, 191,
201 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 ∘ 𝑣) ∈ (𝑅 Cn 𝑇)) |
203 | | imaco 5640 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∘ 𝑣) “ 𝐾) = (𝑢 “ (𝑣 “ 𝐾)) |
204 | | simprrr 805 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))) |
205 | 15 | ntrss2 20861 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑆 ∈ Top ∧ (𝑘‘𝑦) ⊆ ∪ 𝑆) → ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦)) |
206 | 205 | ex 450 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑆 ∈ Top → ((𝑘‘𝑦) ⊆ ∪ 𝑆 → ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦))) |
207 | 206 | ralimdv 2963 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑆 ∈ Top →
(∀𝑦 ∈ 𝑤 (𝑘‘𝑦) ⊆ ∪ 𝑆 → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦))) |
208 | 90, 130, 207 | sylc 65 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦)) |
209 | | ss2iun 4536 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑦 ∈
𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ (𝑘‘𝑦) → ∪
𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) |
210 | 208, 209 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ 𝑦 ∈ 𝑤 (𝑘‘𝑦)) |
211 | 210, 111 | sseqtrd 3641 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ ran
𝑘) |
212 | 211 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦)) ⊆ ∪ ran
𝑘) |
213 | 204, 212 | sstrd 3613 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑣 “ 𝐾) ⊆ ∪ ran
𝑘) |
214 | | imass2 5501 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 “ 𝐾) ⊆ ∪ ran
𝑘 → (𝑢 “ (𝑣 “ 𝐾)) ⊆ (𝑢 “ ∪ ran
𝑘)) |
215 | 213, 214 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 “ (𝑣 “ 𝐾)) ⊆ (𝑢 “ ∪ ran
𝑘)) |
216 | | simprlr 803 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) |
217 | 215, 216 | sstrd 3613 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 “ (𝑣 “ 𝐾)) ⊆ 𝑉) |
218 | 203, 217 | syl5eqss 3649 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → ((𝑢 ∘ 𝑣) “ 𝐾) ⊆ 𝑉) |
219 | | imaeq1 5461 |
. . . . . . . . . . . . . . . 16
⊢ (ℎ = (𝑢 ∘ 𝑣) → (ℎ “ 𝐾) = ((𝑢 ∘ 𝑣) “ 𝐾)) |
220 | 219 | sseq1d 3632 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = (𝑢 ∘ 𝑣) → ((ℎ “ 𝐾) ⊆ 𝑉 ↔ ((𝑢 ∘ 𝑣) “ 𝐾) ⊆ 𝑉)) |
221 | 220 | elrab 3363 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∘ 𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ((𝑢 ∘ 𝑣) ∈ (𝑅 Cn 𝑇) ∧ ((𝑢 ∘ 𝑣) “ 𝐾) ⊆ 𝑉)) |
222 | 202, 218,
221 | sylanbrc 698 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢 ∘ 𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) |
223 | 200, 222 | eqeltrd 2701 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ ((𝑢 ∈ (𝑆 Cn 𝑇) ∧ (𝑢 “ ∪ ran
𝑘) ⊆ 𝑉) ∧ (𝑣 ∈ (𝑅 Cn 𝑆) ∧ (𝑣 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))))) → (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) |
224 | 190, 223 | sylan2b 492 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) ∧ (𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ∧ 𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) → (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) |
225 | 224 | ralrimivva 2971 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) |
226 | 195 | mpt2fun 6762 |
. . . . . . . . . . 11
⊢ Fun 𝐹 |
227 | | ssrab2 3687 |
. . . . . . . . . . . . 13
⊢ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇) |
228 | | ssrab2 3687 |
. . . . . . . . . . . . 13
⊢ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ⊆ (𝑅 Cn 𝑆) |
229 | | xpss12 5225 |
. . . . . . . . . . . . 13
⊢ (({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} ⊆ (𝑆 Cn 𝑇) ∧ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} ⊆ (𝑅 Cn 𝑆)) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆))) |
230 | 227, 228,
229 | mp2an 708 |
. . . . . . . . . . . 12
⊢ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) |
231 | | vex 3203 |
. . . . . . . . . . . . . 14
⊢ 𝑓 ∈ V |
232 | | vex 3203 |
. . . . . . . . . . . . . 14
⊢ 𝑔 ∈ V |
233 | 231, 232 | coex 7118 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∘ 𝑔) ∈ V |
234 | 195, 233 | dmmpt2 7240 |
. . . . . . . . . . . 12
⊢ dom 𝐹 = ((𝑆 Cn 𝑇) × (𝑅 Cn 𝑆)) |
235 | 230, 234 | sseqtr4i 3638 |
. . . . . . . . . . 11
⊢ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ dom 𝐹 |
236 | | funimassov 6811 |
. . . . . . . . . . 11
⊢ ((Fun
𝐹 ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ dom 𝐹) → ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})) |
237 | 226, 235,
236 | mp2an 708 |
. . . . . . . . . 10
⊢ ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ∀𝑢 ∈ {𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉}∀𝑣 ∈ {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))} (𝑢𝐹𝑣) ∈ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) |
238 | 225, 237 | sylibr 224 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → (𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) |
239 | | funimass3 6333 |
. . . . . . . . . 10
⊢ ((Fun
𝐹 ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ dom 𝐹) → ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) |
240 | 226, 235,
239 | mp2an 708 |
. . . . . . . . 9
⊢ ((𝐹 “ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))})) ⊆ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉} ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})) |
241 | 238, 240 | sylib 208 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})) |
242 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → (〈𝐴, 𝐵〉 ∈ 𝑧 ↔ 〈𝐴, 𝐵〉 ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}))) |
243 | | sseq1 3626 |
. . . . . . . . . 10
⊢ (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → (𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}) ↔ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) |
244 | 242, 243 | anbi12d 747 |
. . . . . . . . 9
⊢ (𝑧 = ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) → ((〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})) ↔ (〈𝐴, 𝐵〉 ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))) |
245 | 244 | rspcev 3309 |
. . . . . . . 8
⊢ ((({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) ∧ (〈𝐴, 𝐵〉 ∈ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ∧ ({𝑎 ∈ (𝑆 Cn 𝑇) ∣ (𝑎 “ ∪ ran
𝑘) ⊆ 𝑉} × {𝑏 ∈ (𝑅 Cn 𝑆) ∣ (𝑏 “ 𝐾) ⊆ ∪ 𝑦 ∈ 𝑤 ((int‘𝑆)‘(𝑘‘𝑦))}) ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) |
246 | 139, 183,
241, 245 | syl12anc 1324 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ((𝐵 “ 𝐾) = ∪ 𝑤 ∧ (𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) |
247 | 246 | expr 643 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ (𝐵 “ 𝐾) = ∪ 𝑤) → ((𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))) |
248 | 247 | exlimdv 1861 |
. . . . 5
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ (𝐵 “ 𝐾) = ∪ 𝑤) → (∃𝑘(𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))) |
249 | 89, 248 | syldan 487 |
. . . 4
⊢ (((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) ∧ ∪ (𝑆
↾t (𝐵
“ 𝐾)) = ∪ 𝑤)
→ (∃𝑘(𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp)) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))) |
250 | 249 | expimpd 629 |
. . 3
⊢ ((𝜑 ∧ 𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)) → ((∪ (𝑆
↾t (𝐵
“ 𝐾)) = ∪ 𝑤
∧ ∃𝑘(𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))) |
251 | 250 | rexlimdva 3031 |
. 2
⊢ (𝜑 → (∃𝑤 ∈ (𝒫 (𝑆 ↾t (𝐵 “ 𝐾)) ∩ Fin)(∪
(𝑆 ↾t
(𝐵 “ 𝐾)) = ∪ 𝑤
∧ ∃𝑘(𝑘:𝑤⟶𝒫 (◡𝐴 “ 𝑉) ∧ ∀𝑦 ∈ 𝑤 (𝑦 ⊆ ((int‘𝑆)‘(𝑘‘𝑦)) ∧ (𝑆 ↾t (𝑘‘𝑦)) ∈ Comp))) → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉})))) |
252 | 86, 251 | mpd 15 |
1
⊢ (𝜑 → ∃𝑧 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(〈𝐴, 𝐵〉 ∈ 𝑧 ∧ 𝑧 ⊆ (◡𝐹 “ {ℎ ∈ (𝑅 Cn 𝑇) ∣ (ℎ “ 𝐾) ⊆ 𝑉}))) |