Step | Hyp | Ref
| Expression |
1 | | cnprest.1 |
. . . . . . . 8
⊢ 𝑋 = ∪
𝐽 |
2 | 1 | toptopon 12769 |
. . . . . . 7
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
3 | 2 | biimpi 119 |
. . . . . 6
⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋)) |
4 | 3 | ad2antrr 485 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐽 ∈ (TopOn‘𝑋)) |
5 | 4 | adantr 274 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | | simplr 525 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐾 ∈ Top) |
7 | 6 | adantr 274 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top) |
8 | | simpr 109 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) |
9 | | cnprcl2k 12959 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋) |
10 | 5, 7, 8, 9 | syl3anc 1233 |
. . 3
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝑋) |
11 | 10 | ex 114 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋)) |
12 | 4 | adantr 274 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋)) |
13 | | cnprest.2 |
. . . . . . . . 9
⊢ 𝑌 = ∪
𝐾 |
14 | | uniexg 4422 |
. . . . . . . . 9
⊢ (𝐾 ∈ Top → ∪ 𝐾
∈ V) |
15 | 13, 14 | eqeltrid 2257 |
. . . . . . . 8
⊢ (𝐾 ∈ Top → 𝑌 ∈ V) |
16 | 6, 15 | syl 14 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝑌 ∈ V) |
17 | | simprr 527 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ⊆ 𝑌) |
18 | 16, 17 | ssexd 4127 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐵 ∈ V) |
19 | | resttop 12923 |
. . . . . 6
⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝐾 ↾t 𝐵) ∈ Top) |
20 | 6, 18, 19 | syl2anc 409 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐾 ↾t 𝐵) ∈ Top) |
21 | 20 | adantr 274 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → (𝐾 ↾t 𝐵) ∈ Top) |
22 | | simpr 109 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) |
23 | | cnprcl2k 12959 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾 ↾t 𝐵) ∈ Top ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝑃 ∈ 𝑋) |
24 | 12, 21, 22, 23 | syl3anc 1233 |
. . 3
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)) → 𝑃 ∈ 𝑋) |
25 | 24 | ex 114 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → 𝑃 ∈ 𝑋)) |
26 | | simprl 526 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → 𝐹:𝑋⟶𝐵) |
27 | 26 | ffvelrnda 5628 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝐵) |
28 | 27 | biantrud 302 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹‘𝑃) ∈ 𝑥 ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵))) |
29 | | elin 3310 |
. . . . . . . 8
⊢ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵)) |
30 | 28, 29 | bitr4di 197 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹‘𝑃) ∈ 𝑥 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵))) |
31 | | imassrn 4962 |
. . . . . . . . . . . 12
⊢ (𝐹 “ 𝑦) ⊆ ran 𝐹 |
32 | | simplrl 530 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐹:𝑋⟶𝐵) |
33 | 32 | frnd 5355 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ran 𝐹 ⊆ 𝐵) |
34 | 31, 33 | sstrid 3158 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 “ 𝑦) ⊆ 𝐵) |
35 | 34 | biantrud 302 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ ((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵))) |
36 | | ssin 3349 |
. . . . . . . . . 10
⊢ (((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵) ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)) |
37 | 35, 36 | bitrdi 195 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))) |
38 | 37 | anbi2d 461 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
39 | 38 | rexbidv 2471 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
40 | 30, 39 | imbi12d 233 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))) |
41 | 40 | ralbidv 2470 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))) |
42 | | vex 2733 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
43 | 42 | inex1 4121 |
. . . . . . 7
⊢ (𝑥 ∩ 𝐵) ∈ V |
44 | 43 | a1i 9 |
. . . . . 6
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top) ∧
(𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∩ 𝐵) ∈ V) |
45 | 6 | adantr 274 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ Top) |
46 | 18 | adantr 274 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐵 ∈ V) |
47 | | elrest 12573 |
. . . . . . 7
⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵))) |
48 | 45, 46, 47 | syl2anc 409 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵))) |
49 | | eleq2 2234 |
. . . . . . . 8
⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹‘𝑃) ∈ 𝑧 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵))) |
50 | | sseq2 3171 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹 “ 𝑦) ⊆ 𝑧 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))) |
51 | 50 | anbi2d 461 |
. . . . . . . . 9
⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
52 | 51 | rexbidv 2471 |
. . . . . . . 8
⊢ (𝑧 = (𝑥 ∩ 𝐵) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
53 | 49, 52 | imbi12d 233 |
. . . . . . 7
⊢ (𝑧 = (𝑥 ∩ 𝐵) → (((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))) |
54 | 53 | adantl 275 |
. . . . . 6
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top) ∧
(𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) ∧ 𝑧 = (𝑥 ∩ 𝐵)) → (((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))) |
55 | 44, 48, 54 | ralxfr2d 4447 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))) |
56 | 41, 55 | bitr4d 190 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))) |
57 | 4 | adantr 274 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
58 | 13 | toptopon 12769 |
. . . . . . 7
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
59 | 45, 58 | sylib 121 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
60 | | simpr 109 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋) |
61 | | iscnp 12952 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))) |
62 | 57, 59, 60, 61 | syl3anc 1233 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))) |
63 | 17 | adantr 274 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐵 ⊆ 𝑌) |
64 | 32, 63 | fssd 5358 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝐹:𝑋⟶𝑌) |
65 | 64 | biantrurd 303 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))) |
66 | 62, 65 | bitr4d 190 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))) |
67 | | resttopon 12924 |
. . . . . . 7
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵)) |
68 | 59, 63, 67 | syl2anc 409 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵)) |
69 | | iscnp 12952 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧))))) |
70 | 57, 68, 60, 69 | syl3anc 1233 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧))))) |
71 | 26 | biantrurd 303 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧))))) |
72 | 71 | adantr 274 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)) ↔ (𝐹:𝑋⟶𝐵 ∧ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧))))) |
73 | 70, 72 | bitr4d 190 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))) |
74 | 56, 66, 73 | 3bitr4d 219 |
. . 3
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))) |
75 | 74 | ex 114 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝑃 ∈ 𝑋 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))) |
76 | 11, 25, 75 | pm5.21ndd 700 |
1
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))) |