Step | Hyp | Ref
| Expression |
1 | | cnvimass 4972 |
. . . . . . . 8
⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹 |
2 | | fdm 5351 |
. . . . . . . 8
⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋) |
3 | 1, 2 | sseqtrid 3197 |
. . . . . . 7
⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑦) ⊆ 𝑋) |
4 | 3 | 3ad2ant3 1015 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑦) ⊆ 𝑋) |
5 | 4 | ad2antrr 485 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → (◡𝐹 “ 𝑦) ⊆ 𝑋) |
6 | | neii2 12902 |
. . . . . . . 8
⊢ ((𝐾 ∈ Top ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)) |
7 | 6 | 3ad2antl2 1155 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)) |
8 | 7 | ad2ant2rl 508 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → ∃𝑔 ∈ 𝐾 ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)) |
9 | | cnpnei.1 |
. . . . . . . . . . . 12
⊢ 𝑋 = ∪
𝐽 |
10 | 9 | toptopon 12769 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
11 | 10 | biimpi 119 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋)) |
12 | 11 | 3ad2ant1 1013 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋)) |
13 | 12 | ad3antrrr 489 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋)) |
14 | | cnpnei.2 |
. . . . . . . . . . . 12
⊢ 𝑌 = ∪
𝐾 |
15 | 14 | toptopon 12769 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌)) |
16 | 15 | biimpi 119 |
. . . . . . . . . 10
⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌)) |
17 | 16 | 3ad2ant2 1014 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → 𝐾 ∈ (TopOn‘𝑌)) |
18 | 17 | ad3antrrr 489 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝐾 ∈ (TopOn‘𝑌)) |
19 | | simpllr 529 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝐴 ∈ 𝑋) |
20 | | simplrl 530 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) |
21 | | simprl 526 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → 𝑔 ∈ 𝐾) |
22 | | simprrl 534 |
. . . . . . . . 9
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → {(𝐹‘𝐴)} ⊆ 𝑔) |
23 | | fvexg 5513 |
. . . . . . . . . . 11
⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ V) |
24 | 20, 19, 23 | syl2anc 409 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (𝐹‘𝐴) ∈ V) |
25 | | snssg 3714 |
. . . . . . . . . 10
⊢ ((𝐹‘𝐴) ∈ V → ((𝐹‘𝐴) ∈ 𝑔 ↔ {(𝐹‘𝐴)} ⊆ 𝑔)) |
26 | 24, 25 | syl 14 |
. . . . . . . . 9
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → ((𝐹‘𝐴) ∈ 𝑔 ↔ {(𝐹‘𝐴)} ⊆ 𝑔)) |
27 | 22, 26 | mpbird 166 |
. . . . . . . 8
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (𝐹‘𝐴) ∈ 𝑔) |
28 | | icnpimaex 12964 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑔 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑔)) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔)) |
29 | 13, 18, 19, 20, 21, 27, 28 | syl33anc 1248 |
. . . . . . 7
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔)) |
30 | | sstr2 3154 |
. . . . . . . . . . . . 13
⊢ ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝑔 ⊆ 𝑦 → (𝐹 “ 𝑜) ⊆ 𝑦)) |
31 | 30 | com12 30 |
. . . . . . . . . . . 12
⊢ (𝑔 ⊆ 𝑦 → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦)) |
32 | 31 | ad2antll 488 |
. . . . . . . . . . 11
⊢ ((𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)) → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦)) |
33 | 32 | ad2antlr 486 |
. . . . . . . . . 10
⊢
((((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐹 “ 𝑜) ⊆ 𝑔 → (𝐹 “ 𝑜) ⊆ 𝑦)) |
34 | | ffun 5348 |
. . . . . . . . . . . . . 14
⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹) |
35 | 34 | 3ad2ant3 1015 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → Fun 𝐹) |
36 | 35 | ad2antrr 485 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → Fun 𝐹) |
37 | 36 | ad2antrr 485 |
. . . . . . . . . . 11
⊢
((((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → Fun 𝐹) |
38 | 9 | eltopss 12760 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋) |
39 | 38 | adantlr 474 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ 𝑋) |
40 | 2 | sseq2d 3177 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹:𝑋⟶𝑌 → (𝑜 ⊆ dom 𝐹 ↔ 𝑜 ⊆ 𝑋)) |
41 | 40 | ad2antlr 486 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → (𝑜 ⊆ dom 𝐹 ↔ 𝑜 ⊆ 𝑋)) |
42 | 39, 41 | mpbird 166 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹) |
43 | 42 | 3adantl2 1149 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹) |
44 | 43 | adantlr 474 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹) |
45 | 44 | adantlr 474 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹) |
46 | 45 | adantlr 474 |
. . . . . . . . . . 11
⊢
((((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → 𝑜 ⊆ dom 𝐹) |
47 | | funimass3 5609 |
. . . . . . . . . . 11
⊢ ((Fun
𝐹 ∧ 𝑜 ⊆ dom 𝐹) → ((𝐹 “ 𝑜) ⊆ 𝑦 ↔ 𝑜 ⊆ (◡𝐹 “ 𝑦))) |
48 | 37, 46, 47 | syl2anc 409 |
. . . . . . . . . 10
⊢
((((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐹 “ 𝑜) ⊆ 𝑦 ↔ 𝑜 ⊆ (◡𝐹 “ 𝑦))) |
49 | 33, 48 | sylibd 148 |
. . . . . . . . 9
⊢
((((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐹 “ 𝑜) ⊆ 𝑔 → 𝑜 ⊆ (◡𝐹 “ 𝑦))) |
50 | 49 | anim2d 335 |
. . . . . . . 8
⊢
((((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) ∧ 𝑜 ∈ 𝐽) → ((𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔) → (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦)))) |
51 | 50 | reximdva 2572 |
. . . . . . 7
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → (∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ (𝐹 “ 𝑜) ⊆ 𝑔) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦)))) |
52 | 29, 51 | mpd 13 |
. . . . . 6
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) ∧ (𝑔 ∈ 𝐾 ∧ ({(𝐹‘𝐴)} ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦))) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦))) |
53 | 8, 52 | rexlimddv 2592 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦))) |
54 | 9 | isneip 12899 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → ((◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦))))) |
55 | 54 | 3ad2antl1 1154 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → ((◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦))))) |
56 | 55 | adantr 274 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → ((◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ ((◡𝐹 “ 𝑦) ⊆ 𝑋 ∧ ∃𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ∧ 𝑜 ⊆ (◡𝐹 “ 𝑦))))) |
57 | 5, 53, 56 | mpbir2and 939 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ 𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}))) → (◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) |
58 | 57 | exp32 363 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}) → (◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})))) |
59 | 58 | ralrimdv 2549 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}))) |
60 | | simpll3 1033 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹:𝑋⟶𝑌) |
61 | | opnneip 12912 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Top ∧ 𝑜 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})) |
62 | | imaeq2 4947 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑜 → (◡𝐹 “ 𝑦) = (◡𝐹 “ 𝑜)) |
63 | 62 | eleq1d 2239 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑜 → ((◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) ↔ (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
64 | 63 | rspcv 2830 |
. . . . . . . . . . . . . 14
⊢ (𝑜 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)}) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
65 | 61, 64 | syl 14 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Top ∧ 𝑜 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑜) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
66 | 65 | 3com23 1204 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Top ∧ (𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
67 | 66 | 3expb 1199 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ Top ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
68 | 67 | 3ad2antl2 1155 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
69 | 68 | adantlr 474 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}))) |
70 | | neii2 12902 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ (◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜))) |
71 | 70 | ex 114 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ Top → ((◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜)))) |
72 | 71 | 3ad2ant1 1013 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → ((◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜)))) |
73 | 72 | ad2antrr 485 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → ((◡𝐹 “ 𝑜) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜)))) |
74 | | snssg 3714 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑔 ↔ {𝐴} ⊆ 𝑔)) |
75 | 74 | ad3antlr 490 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → (𝐴 ∈ 𝑔 ↔ {𝐴} ⊆ 𝑔)) |
76 | 35 | ad3antrrr 489 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → Fun 𝐹) |
77 | 9 | eltopss 12760 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ 𝑋) |
78 | 77 | 3ad2antl1 1154 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ 𝑋) |
79 | 2 | sseq2d 3177 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹:𝑋⟶𝑌 → (𝑔 ⊆ dom 𝐹 ↔ 𝑔 ⊆ 𝑋)) |
80 | 79 | 3ad2ant3 1015 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) → (𝑔 ⊆ dom 𝐹 ↔ 𝑔 ⊆ 𝑋)) |
81 | 80 | biimpar 295 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ⊆ 𝑋) → 𝑔 ⊆ dom 𝐹) |
82 | 78, 81 | syldan 280 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹) |
83 | 82 | adantlr 474 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹) |
84 | 83 | adantlr 474 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → 𝑔 ⊆ dom 𝐹) |
85 | | funimass3 5609 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐹 ∧ 𝑔 ⊆ dom 𝐹) → ((𝐹 “ 𝑔) ⊆ 𝑜 ↔ 𝑔 ⊆ (◡𝐹 “ 𝑜))) |
86 | 76, 84, 85 | syl2anc 409 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → ((𝐹 “ 𝑔) ⊆ 𝑜 ↔ 𝑔 ⊆ (◡𝐹 “ 𝑜))) |
87 | 75, 86 | anbi12d 470 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → ((𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜) ↔ ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜)))) |
88 | 87 | biimprd 157 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ Top
∧ 𝐾 ∈ Top ∧
𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) ∧ 𝑔 ∈ 𝐽) → (({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜)) → (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))) |
89 | 88 | reximdva 2572 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∃𝑔 ∈ 𝐽 ({𝐴} ⊆ 𝑔 ∧ 𝑔 ⊆ (◡𝐹 “ 𝑜)) → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))) |
90 | 69, 73, 89 | 3syld 57 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ((𝐹‘𝐴) ∈ 𝑜 ∧ 𝑜 ∈ 𝐾)) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))) |
91 | 90 | exp32 363 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ∈ 𝑜 → (𝑜 ∈ 𝐾 → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
92 | 91 | com24 87 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → (𝑜 ∈ 𝐾 → ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
93 | 92 | imp 123 |
. . . . 5
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → (𝑜 ∈ 𝐾 → ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜)))) |
94 | 93 | ralrimiv 2542 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))) |
95 | 11 | 3ad2ant1 1013 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
96 | 16 | 3ad2ant2 1014 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
97 | | simp3 994 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋) |
98 | | iscnp 12952 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
99 | 95, 96, 97, 98 | syl3anc 1233 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
100 | 99 | 3expa 1198 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
101 | 100 | 3adantl3 1150 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
102 | 101 | adantr 274 |
. . . 4
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑜 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑜 → ∃𝑔 ∈ 𝐽 (𝐴 ∈ 𝑔 ∧ (𝐹 “ 𝑔) ⊆ 𝑜))))) |
103 | 60, 94, 102 | mpbir2and 939 |
. . 3
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴})) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) |
104 | 103 | ex 114 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))) |
105 | 59, 104 | impbid 128 |
1
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹:𝑋⟶𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ ∀𝑦 ∈ ((nei‘𝐾)‘{(𝐹‘𝐴)})(◡𝐹 “ 𝑦) ∈ ((nei‘𝐽)‘{𝐴}))) |