Step | Hyp | Ref
| Expression |
1 | | cnptop1 22393 |
. . . 4
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top) |
2 | | cnprest.1 |
. . . . 5
⊢ 𝑋 = ∪
𝐽 |
3 | 2 | cnprcl 22396 |
. . . 4
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ 𝑋) |
4 | 1, 3 | jca 512 |
. . 3
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) |
5 | 4 | a1i 11 |
. 2
⊢ ((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋))) |
6 | | cnptop1 22393 |
. . . 4
⊢ (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → 𝐽 ∈ Top) |
7 | 2 | cnprcl 22396 |
. . . 4
⊢ (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → 𝑃 ∈ 𝑋) |
8 | 6, 7 | jca 512 |
. . 3
⊢ (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) |
9 | 8 | a1i 11 |
. 2
⊢ ((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) → (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋))) |
10 | | simpl2 1191 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐹:𝑋⟶𝐵) |
11 | | simprr 770 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝑃 ∈ 𝑋) |
12 | 10, 11 | ffvelrnd 6962 |
. . . . . . . . 9
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐹‘𝑃) ∈ 𝐵) |
13 | 12 | biantrud 532 |
. . . . . . . 8
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → ((𝐹‘𝑃) ∈ 𝑥 ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵))) |
14 | | elin 3903 |
. . . . . . . 8
⊢ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) ↔ ((𝐹‘𝑃) ∈ 𝑥 ∧ (𝐹‘𝑃) ∈ 𝐵)) |
15 | 13, 14 | bitr4di 289 |
. . . . . . 7
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → ((𝐹‘𝑃) ∈ 𝑥 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵))) |
16 | | imassrn 5980 |
. . . . . . . . . . . 12
⊢ (𝐹 “ 𝑦) ⊆ ran 𝐹 |
17 | 10 | frnd 6608 |
. . . . . . . . . . . 12
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → ran 𝐹 ⊆ 𝐵) |
18 | 16, 17 | sstrid 3932 |
. . . . . . . . . . 11
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐹 “ 𝑦) ⊆ 𝐵) |
19 | 18 | biantrud 532 |
. . . . . . . . . 10
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ ((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵))) |
20 | | ssin 4164 |
. . . . . . . . . 10
⊢ (((𝐹 “ 𝑦) ⊆ 𝑥 ∧ (𝐹 “ 𝑦) ⊆ 𝐵) ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)) |
21 | 19, 20 | bitrdi 287 |
. . . . . . . . 9
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → ((𝐹 “ 𝑦) ⊆ 𝑥 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))) |
22 | 21 | anbi2d 629 |
. . . . . . . 8
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
23 | 22 | rexbidv 3226 |
. . . . . . 7
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
24 | 15, 23 | imbi12d 345 |
. . . . . 6
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)) ↔ ((𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵) → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))))) |
25 | 24 | ralbidv 3112 |
. . . . 5
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 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))))) |
26 | | vex 3436 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
27 | 26 | inex1 5241 |
. . . . . . 7
⊢ (𝑥 ∩ 𝐵) ∈ V |
28 | 27 | a1i 11 |
. . . . . 6
⊢ ((((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) ∧ 𝑥 ∈ 𝐾) → (𝑥 ∩ 𝐵) ∈ V) |
29 | | simpl1 1190 |
. . . . . . 7
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐾 ∈ Top) |
30 | | cnprest.2 |
. . . . . . . . . 10
⊢ 𝑌 = ∪
𝐾 |
31 | | uniexg 7593 |
. . . . . . . . . 10
⊢ (𝐾 ∈ Top → ∪ 𝐾
∈ V) |
32 | 30, 31 | eqeltrid 2843 |
. . . . . . . . 9
⊢ (𝐾 ∈ Top → 𝑌 ∈ V) |
33 | 29, 32 | syl 17 |
. . . . . . . 8
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝑌 ∈ V) |
34 | | simpl3 1192 |
. . . . . . . 8
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐵 ⊆ 𝑌) |
35 | 33, 34 | ssexd 5248 |
. . . . . . 7
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐵 ∈ V) |
36 | | elrest 17138 |
. . . . . . 7
⊢ ((𝐾 ∈ Top ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵))) |
37 | 29, 35, 36 | syl2anc 584 |
. . . . . 6
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝑧 ∈ (𝐾 ↾t 𝐵) ↔ ∃𝑥 ∈ 𝐾 𝑧 = (𝑥 ∩ 𝐵))) |
38 | | eleq2 2827 |
. . . . . . . 8
⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹‘𝑃) ∈ 𝑧 ↔ (𝐹‘𝑃) ∈ (𝑥 ∩ 𝐵))) |
39 | | sseq2 3947 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝐹 “ 𝑦) ⊆ 𝑧 ↔ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵))) |
40 | 39 | anbi2d 629 |
. . . . . . . . 9
⊢ (𝑧 = (𝑥 ∩ 𝐵) → ((𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
41 | 40 | rexbidv 3226 |
. . . . . . . 8
⊢ (𝑧 = (𝑥 ∩ 𝐵) → (∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧) ↔ ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ (𝑥 ∩ 𝐵)))) |
42 | 38, 41 | imbi12d 345 |
. . . . . . 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))))) |
43 | 42 | adantl 482 |
. . . . . 6
⊢ ((((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 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))))) |
44 | 28, 37, 43 | ralxfr2d 5333 |
. . . . 5
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 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))))) |
45 | 25, 44 | bitr4d 281 |
. . . 4
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 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)))) |
46 | 10, 34 | fssd 6618 |
. . . . 5
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌) |
47 | | simprl 768 |
. . . . . 6
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐽 ∈ Top) |
48 | 2, 30 | iscnp2 22390 |
. . . . . . 7
⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))) |
49 | 48 | baib 536 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))) |
50 | 47, 29, 11, 49 | syl3anc 1370 |
. . . . 5
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥))))) |
51 | 46, 50 | mpbirand 704 |
. . . 4
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ ∀𝑥 ∈ 𝐾 ((𝐹‘𝑃) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑥)))) |
52 | 2 | toptopon 22066 |
. . . . . . 7
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
53 | 47, 52 | sylib 217 |
. . . . . 6
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋)) |
54 | 30 | toptopon 22066 |
. . . . . . . 8
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
55 | 29, 54 | sylib 217 |
. . . . . . 7
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘𝑌)) |
56 | | resttopon 22312 |
. . . . . . 7
⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ⊆ 𝑌) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵)) |
57 | 55, 34, 56 | syl2anc 584 |
. . . . . 6
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐾 ↾t 𝐵) ∈ (TopOn‘𝐵)) |
58 | | iscnp 22388 |
. . . . . 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))))) |
59 | 53, 57, 11, 58 | syl3anc 1370 |
. . . . 5
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 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))))) |
60 | 10, 59 | mpbirand 704 |
. . . 4
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃) ↔ ∀𝑧 ∈ (𝐾 ↾t 𝐵)((𝐹‘𝑃) ∈ 𝑧 → ∃𝑦 ∈ 𝐽 (𝑃 ∈ 𝑦 ∧ (𝐹 “ 𝑦) ⊆ 𝑧)))) |
61 | 45, 51, 60 | 3bitr4d 311 |
. . 3
⊢ (((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))) |
62 | 61 | ex 413 |
. 2
⊢ ((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) → ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃)))) |
63 | 5, 9, 62 | pm5.21ndd 381 |
1
⊢ ((𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝐵 ∧ 𝐵 ⊆ 𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ 𝐹 ∈ ((𝐽 CnP (𝐾 ↾t 𝐵))‘𝑃))) |