Step | Hyp | Ref
| Expression |
1 | | neibastop1.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
2 | | neibastop1.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖
{∅})) |
3 | | neibastop1.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) |
4 | | neibastop1.4 |
. . . . . . . . 9
⊢ 𝐽 = {𝑜 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅} |
5 | 1, 2, 3, 4 | neibastop1 32942 |
. . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
6 | | topontop 21125 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
7 | 5, 6 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐽 ∈ Top) |
8 | 7 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑃 ∈ 𝑋) → 𝐽 ∈ Top) |
9 | | eqid 2777 |
. . . . . . 7
⊢ ∪ 𝐽 =
∪ 𝐽 |
10 | 9 | neii1 21318 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑁 ⊆ ∪ 𝐽) |
11 | 8, 10 | sylan 575 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑁 ⊆ ∪ 𝐽) |
12 | | toponuni 21126 |
. . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
13 | 5, 12 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
14 | 13 | ad2antrr 716 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑋 = ∪ 𝐽) |
15 | 11, 14 | sseqtr4d 3860 |
. . . 4
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → 𝑁 ⊆ 𝑋) |
16 | | neii2 21320 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∃𝑦 ∈ 𝐽 ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁)) |
17 | 8, 16 | sylan 575 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∃𝑦 ∈ 𝐽 ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁)) |
18 | | pweq 4381 |
. . . . . . . . . . 11
⊢ (𝑜 = 𝑦 → 𝒫 𝑜 = 𝒫 𝑦) |
19 | 18 | ineq2d 4036 |
. . . . . . . . . 10
⊢ (𝑜 = 𝑦 → ((𝐹‘𝑥) ∩ 𝒫 𝑜) = ((𝐹‘𝑥) ∩ 𝒫 𝑦)) |
20 | 19 | neeq1d 3027 |
. . . . . . . . 9
⊢ (𝑜 = 𝑦 → (((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅ ↔ ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅)) |
21 | 20 | raleqbi1dv 3327 |
. . . . . . . 8
⊢ (𝑜 = 𝑦 → (∀𝑥 ∈ 𝑜 ((𝐹‘𝑥) ∩ 𝒫 𝑜) ≠ ∅ ↔ ∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅)) |
22 | 21, 4 | elrab2 3575 |
. . . . . . 7
⊢ (𝑦 ∈ 𝐽 ↔ (𝑦 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅)) |
23 | | simprrr 772 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → 𝑦 ⊆ 𝑁) |
24 | | sspwb 5149 |
. . . . . . . . . . . . 13
⊢ (𝑦 ⊆ 𝑁 ↔ 𝒫 𝑦 ⊆ 𝒫 𝑁) |
25 | 23, 24 | sylib 210 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → 𝒫 𝑦 ⊆ 𝒫 𝑁) |
26 | | sslin 4058 |
. . . . . . . . . . . 12
⊢
(𝒫 𝑦 ⊆
𝒫 𝑁 → ((𝐹‘𝑃) ∩ 𝒫 𝑦) ⊆ ((𝐹‘𝑃) ∩ 𝒫 𝑁)) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → ((𝐹‘𝑃) ∩ 𝒫 𝑦) ⊆ ((𝐹‘𝑃) ∩ 𝒫 𝑁)) |
28 | | simprrl 771 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → {𝑃} ⊆ 𝑦) |
29 | | snssg 4547 |
. . . . . . . . . . . . . 14
⊢ (𝑃 ∈ 𝑋 → (𝑃 ∈ 𝑦 ↔ {𝑃} ⊆ 𝑦)) |
30 | 29 | ad3antlr 721 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → (𝑃 ∈ 𝑦 ↔ {𝑃} ⊆ 𝑦)) |
31 | 28, 30 | mpbird 249 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → 𝑃 ∈ 𝑦) |
32 | | fveq2 6446 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑃 → (𝐹‘𝑥) = (𝐹‘𝑃)) |
33 | 32 | ineq1d 4035 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑃 → ((𝐹‘𝑥) ∩ 𝒫 𝑦) = ((𝐹‘𝑃) ∩ 𝒫 𝑦)) |
34 | 33 | neeq1d 3027 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑃 → (((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅ ↔ ((𝐹‘𝑃) ∩ 𝒫 𝑦) ≠ ∅)) |
35 | 34 | rspcv 3506 |
. . . . . . . . . . . 12
⊢ (𝑃 ∈ 𝑦 → (∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅ → ((𝐹‘𝑃) ∩ 𝒫 𝑦) ≠ ∅)) |
36 | 31, 35 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → (∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅ → ((𝐹‘𝑃) ∩ 𝒫 𝑦) ≠ ∅)) |
37 | | ssn0 4201 |
. . . . . . . . . . 11
⊢ ((((𝐹‘𝑃) ∩ 𝒫 𝑦) ⊆ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑦) ≠ ∅) → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅) |
38 | 27, 36, 37 | syl6an 674 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑦 ∈ 𝒫 𝑋 ∧ ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁))) → (∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅ → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅)) |
39 | 38 | expr 450 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑦 ∈ 𝒫 𝑋) → (({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁) → (∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅ → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |
40 | 39 | com23 86 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑦 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅ → (({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁) → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |
41 | 40 | expimpd 447 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ((𝑦 ∈ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝑦 ((𝐹‘𝑥) ∩ 𝒫 𝑦) ≠ ∅) → (({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁) → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |
42 | 22, 41 | syl5bi 234 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑦 ∈ 𝐽 → (({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁) → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |
43 | 42 | rexlimdv 3211 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (∃𝑦 ∈ 𝐽 ({𝑃} ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑁) → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅)) |
44 | 17, 43 | mpd 15 |
. . . 4
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅) |
45 | 15, 44 | jca 507 |
. . 3
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑁 ⊆ 𝑋 ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅)) |
46 | 45 | ex 403 |
. 2
⊢ ((𝜑 ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) → (𝑁 ⊆ 𝑋 ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |
47 | | n0 4158 |
. . . 4
⊢ (((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ ((𝐹‘𝑃) ∩ 𝒫 𝑁)) |
48 | | elin 4018 |
. . . . . 6
⊢ (𝑠 ∈ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ↔ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁)) |
49 | | simprl 761 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑁 ⊆ 𝑋) |
50 | 13 | ad2antrr 716 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑋 = ∪ 𝐽) |
51 | 49, 50 | sseqtrd 3859 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑁 ⊆ ∪ 𝐽) |
52 | 1 | ad2antrr 716 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑋 ∈ 𝑉) |
53 | 2 | ad2antrr 716 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝐹:𝑋⟶(𝒫 𝒫 𝑋 ∖
{∅})) |
54 | | simpll 757 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝜑) |
55 | 54, 3 | sylan 575 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ (𝐹‘𝑥))) → ((𝐹‘𝑥) ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅) |
56 | | neibastop1.5 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) |
57 | 54, 56 | sylan 575 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → 𝑥 ∈ 𝑣) |
58 | | neibastop1.6 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) |
59 | 54, 58 | sylan 575 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑣 ∈ (𝐹‘𝑥))) → ∃𝑡 ∈ (𝐹‘𝑥)∀𝑦 ∈ 𝑡 ((𝐹‘𝑦) ∩ 𝒫 𝑣) ≠ ∅) |
60 | | simplr 759 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑃 ∈ 𝑋) |
61 | | simprrl 771 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑠 ∈ (𝐹‘𝑃)) |
62 | | simprrr 772 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑠 ∈ 𝒫 𝑁) |
63 | 62 | elpwid 4390 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑠 ⊆ 𝑁) |
64 | | fveq2 6446 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑥 → (𝐹‘𝑛) = (𝐹‘𝑥)) |
65 | 64 | ineq1d 4035 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑥 → ((𝐹‘𝑛) ∩ 𝒫 𝑏) = ((𝐹‘𝑥) ∩ 𝒫 𝑏)) |
66 | 65 | cbviunv 4792 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏) = ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑏) |
67 | | pweq 4381 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑧 → 𝒫 𝑏 = 𝒫 𝑧) |
68 | 67 | ineq2d 4036 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑧 → ((𝐹‘𝑥) ∩ 𝒫 𝑏) = ((𝐹‘𝑥) ∩ 𝒫 𝑧)) |
69 | 68 | iuneq2d 4780 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = 𝑧 → ∪
𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑏) = ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)) |
70 | 66, 69 | syl5eq 2825 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑧 → ∪
𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏) = ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)) |
71 | 70 | cbviunv 4792 |
. . . . . . . . . . . 12
⊢ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏) = ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧) |
72 | 71 | mpteq2i 4976 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)) = (𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)) |
73 | | rdgeq1 7790 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)) = (𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)) → rec((𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) = rec((𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)), {𝑠})) |
74 | 72, 73 | ax-mp 5 |
. . . . . . . . . 10
⊢
rec((𝑎 ∈ V
↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) = rec((𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)), {𝑠}) |
75 | 74 | reseq1i 5638 |
. . . . . . . . 9
⊢
(rec((𝑎 ∈ V
↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑧 ∈ 𝑎 ∪ 𝑥 ∈ 𝑋 ((𝐹‘𝑥) ∩ 𝒫 𝑧)), {𝑠}) ↾ ω) |
76 | | pweq 4381 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝑓 → 𝒫 𝑔 = 𝒫 𝑓) |
77 | 76 | ineq2d 4036 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑓 → ((𝐹‘𝑤) ∩ 𝒫 𝑔) = ((𝐹‘𝑤) ∩ 𝒫 𝑓)) |
78 | 77 | neeq1d 3027 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑓 → (((𝐹‘𝑤) ∩ 𝒫 𝑔) ≠ ∅ ↔ ((𝐹‘𝑤) ∩ 𝒫 𝑓) ≠ ∅)) |
79 | 78 | cbvrexv 3367 |
. . . . . . . . . . 11
⊢
(∃𝑔 ∈
∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑤) ∩ 𝒫 𝑔) ≠ ∅ ↔ ∃𝑓 ∈ ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑤) ∩ 𝒫 𝑓) ≠ ∅) |
80 | | fveq2 6446 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑦 → (𝐹‘𝑤) = (𝐹‘𝑦)) |
81 | 80 | ineq1d 4035 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑦 → ((𝐹‘𝑤) ∩ 𝒫 𝑓) = ((𝐹‘𝑦) ∩ 𝒫 𝑓)) |
82 | 81 | neeq1d 3027 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑦 → (((𝐹‘𝑤) ∩ 𝒫 𝑓) ≠ ∅ ↔ ((𝐹‘𝑦) ∩ 𝒫 𝑓) ≠ ∅)) |
83 | 82 | rexbidv 3236 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑦 → (∃𝑓 ∈ ∪ ran
(rec((𝑎 ∈ V ↦
∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑤) ∩ 𝒫 𝑓) ≠ ∅ ↔ ∃𝑓 ∈ ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑦) ∩ 𝒫 𝑓) ≠ ∅)) |
84 | 79, 83 | syl5bb 275 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → (∃𝑔 ∈ ∪ ran
(rec((𝑎 ∈ V ↦
∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑤) ∩ 𝒫 𝑔) ≠ ∅ ↔ ∃𝑓 ∈ ∪ ran (rec((𝑎 ∈ V ↦ ∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑦) ∩ 𝒫 𝑓) ≠ ∅)) |
85 | 84 | cbvrabv 3395 |
. . . . . . . . 9
⊢ {𝑤 ∈ 𝑋 ∣ ∃𝑔 ∈ ∪ ran
(rec((𝑎 ∈ V ↦
∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑤) ∩ 𝒫 𝑔) ≠ ∅} = {𝑦 ∈ 𝑋 ∣ ∃𝑓 ∈ ∪ ran
(rec((𝑎 ∈ V ↦
∪ 𝑏 ∈ 𝑎 ∪ 𝑛 ∈ 𝑋 ((𝐹‘𝑛) ∩ 𝒫 𝑏)), {𝑠}) ↾ ω)((𝐹‘𝑦) ∩ 𝒫 𝑓) ≠ ∅} |
86 | 52, 53, 55, 4, 57, 59, 60, 49, 61, 63, 75, 85 | neibastop2lem 32943 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑁)) |
87 | 7 | ad2antrr 716 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝐽 ∈ Top) |
88 | 60, 50 | eleqtrd 2860 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑃 ∈ ∪ 𝐽) |
89 | 9 | isneip 21317 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ ∪ 𝐽)
→ (𝑁 ∈
((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ ∪ 𝐽 ∧ ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑁)))) |
90 | 87, 88, 89 | syl2anc 579 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ ∪ 𝐽 ∧ ∃𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑁)))) |
91 | 51, 86, 90 | mpbir2and 703 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ (𝑁 ⊆ 𝑋 ∧ (𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁))) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) |
92 | 91 | expr 450 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ⊆ 𝑋) → ((𝑠 ∈ (𝐹‘𝑃) ∧ 𝑠 ∈ 𝒫 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) |
93 | 48, 92 | syl5bi 234 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ⊆ 𝑋) → (𝑠 ∈ ((𝐹‘𝑃) ∩ 𝒫 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) |
94 | 93 | exlimdv 1976 |
. . . 4
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ⊆ 𝑋) → (∃𝑠 𝑠 ∈ ((𝐹‘𝑃) ∩ 𝒫 𝑁) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) |
95 | 47, 94 | syl5bi 234 |
. . 3
⊢ (((𝜑 ∧ 𝑃 ∈ 𝑋) ∧ 𝑁 ⊆ 𝑋) → (((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅ → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) |
96 | 95 | expimpd 447 |
. 2
⊢ ((𝜑 ∧ 𝑃 ∈ 𝑋) → ((𝑁 ⊆ 𝑋 ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅) → 𝑁 ∈ ((nei‘𝐽)‘{𝑃}))) |
97 | 46, 96 | impbid 204 |
1
⊢ ((𝜑 ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ((𝐹‘𝑃) ∩ 𝒫 𝑁) ≠ ∅))) |