Step | Hyp | Ref
| Expression |
1 | | 1stcclb.1 |
. . 3
⊢ 𝑋 = ∪
𝐽 |
2 | 1 | 1stcclb 21247 |
. 2
⊢ ((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) |
3 | | 1stctop 21246 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ 1st𝜔
→ 𝐽 ∈
Top) |
4 | 3 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝐽 ∈ Top) |
5 | 1 | topopn 20711 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
6 | 4, 5 | syl 17 |
. . . . . . . . 9
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝑋 ∈ 𝐽) |
7 | | simprrr 805 |
. . . . . . . . 9
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) |
8 | | simplr 792 |
. . . . . . . . 9
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝐴 ∈ 𝑋) |
9 | | eleq2 2690 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑋 → (𝐴 ∈ 𝑧 ↔ 𝐴 ∈ 𝑋)) |
10 | | sseq2 3627 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑋 → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ 𝑋)) |
11 | 10 | anbi2d 740 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑋 → ((𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋))) |
12 | 11 | rexbidv 3052 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑋 → (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋))) |
13 | 9, 12 | imbi12d 334 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑋 → ((𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝐴 ∈ 𝑋 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋)))) |
14 | 13 | rspcv 3305 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐽 → (∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) → (𝐴 ∈ 𝑋 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋)))) |
15 | 6, 7, 8, 14 | syl3c 66 |
. . . . . . . 8
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋)) |
16 | | simpl 473 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋) → 𝐴 ∈ 𝑤) |
17 | 16 | reximi 3011 |
. . . . . . . 8
⊢
(∃𝑤 ∈
𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑋) → ∃𝑤 ∈ 𝑥 𝐴 ∈ 𝑤) |
18 | 15, 17 | syl 17 |
. . . . . . 7
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑤 ∈ 𝑥 𝐴 ∈ 𝑤) |
19 | | eleq2 2690 |
. . . . . . . 8
⊢ (𝑤 = 𝑎 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑎)) |
20 | 19 | cbvrexv 3172 |
. . . . . . 7
⊢
(∃𝑤 ∈
𝑥 𝐴 ∈ 𝑤 ↔ ∃𝑎 ∈ 𝑥 𝐴 ∈ 𝑎) |
21 | 18, 20 | sylib 208 |
. . . . . 6
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑎 ∈ 𝑥 𝐴 ∈ 𝑎) |
22 | | rabn0 3958 |
. . . . . 6
⊢ ({𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ 𝑥 𝐴 ∈ 𝑎) |
23 | 21, 22 | sylibr 224 |
. . . . 5
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≠ ∅) |
24 | | vex 3203 |
. . . . . . 7
⊢ 𝑥 ∈ V |
25 | 24 | rabex 4813 |
. . . . . 6
⊢ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ∈ V |
26 | 25 | 0sdom 8091 |
. . . . 5
⊢ (∅
≺ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ↔ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≠ ∅) |
27 | 23, 26 | sylibr 224 |
. . . 4
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∅ ≺ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
28 | | ssrab2 3687 |
. . . . . 6
⊢ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ⊆ 𝑥 |
29 | | ssdomg 8001 |
. . . . . 6
⊢ (𝑥 ∈ V → ({𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ⊆ 𝑥 → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ 𝑥)) |
30 | 24, 28, 29 | mp2 9 |
. . . . 5
⊢ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ 𝑥 |
31 | | simprrl 804 |
. . . . . 6
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝑥 ≼ ω) |
32 | | nnenom 12779 |
. . . . . . 7
⊢ ℕ
≈ ω |
33 | 32 | ensymi 8006 |
. . . . . 6
⊢ ω
≈ ℕ |
34 | | domentr 8015 |
. . . . . 6
⊢ ((𝑥 ≼ ω ∧ ω
≈ ℕ) → 𝑥
≼ ℕ) |
35 | 31, 33, 34 | sylancl 694 |
. . . . 5
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → 𝑥 ≼ ℕ) |
36 | | domtr 8009 |
. . . . 5
⊢ (({𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ 𝑥 ∧ 𝑥 ≼ ℕ) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ ℕ) |
37 | 30, 35, 36 | sylancr 695 |
. . . 4
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ ℕ) |
38 | | fodomr 8111 |
. . . 4
⊢ ((∅
≺ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ∧ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ≼ ℕ) → ∃𝑔 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
39 | 27, 37, 38 | syl2anc 693 |
. . 3
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑔 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
40 | 3 | ad3antrrr 766 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Top) |
41 | | imassrn 5477 |
. . . . . . . . . 10
⊢ (𝑔 “ (1...𝑛)) ⊆ ran 𝑔 |
42 | | forn 6118 |
. . . . . . . . . . . . 13
⊢ (𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ran 𝑔 = {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
43 | 42 | ad2antll 765 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ran 𝑔 = {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
44 | | simprll 802 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑥 ∈ 𝒫 𝐽) |
45 | 44 | elpwid 4170 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑥 ⊆ 𝐽) |
46 | 28, 45 | syl5ss 3614 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} ⊆ 𝐽) |
47 | 43, 46 | eqsstrd 3639 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ran 𝑔 ⊆ 𝐽) |
48 | 47 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → ran 𝑔 ⊆ 𝐽) |
49 | 41, 48 | syl5ss 3614 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ⊆ 𝐽) |
50 | | elfznn 12370 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (1...𝑛) → 𝑘 ∈ ℕ) |
51 | 50 | ssriv 3607 |
. . . . . . . . . . . . . 14
⊢
(1...𝑛) ⊆
ℕ |
52 | | fof 6115 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → 𝑔:ℕ⟶{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
53 | 52 | ad2antll 765 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑔:ℕ⟶{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
54 | | fdm 6051 |
. . . . . . . . . . . . . . 15
⊢ (𝑔:ℕ⟶{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → dom 𝑔 = ℕ) |
55 | 53, 54 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → dom 𝑔 = ℕ) |
56 | 51, 55 | syl5sseqr 3654 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (1...𝑛) ⊆ dom 𝑔) |
57 | 56 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ dom 𝑔) |
58 | | sseqin2 3817 |
. . . . . . . . . . . 12
⊢
((1...𝑛) ⊆ dom
𝑔 ↔ (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛)) |
59 | 57, 58 | sylib 208 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) = (1...𝑛)) |
60 | | elfz1end 12371 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (1...𝑛)) |
61 | | ne0i 3921 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (1...𝑛) → (1...𝑛) ≠ ∅) |
62 | 61 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ (1...𝑛)) → (1...𝑛) ≠ ∅) |
63 | 60, 62 | sylan2b 492 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ≠ ∅) |
64 | 59, 63 | eqnetrd 2861 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (dom 𝑔 ∩ (1...𝑛)) ≠ ∅) |
65 | | imadisj 5484 |
. . . . . . . . . . 11
⊢ ((𝑔 “ (1...𝑛)) = ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) = ∅) |
66 | 65 | necon3bii 2846 |
. . . . . . . . . 10
⊢ ((𝑔 “ (1...𝑛)) ≠ ∅ ↔ (dom 𝑔 ∩ (1...𝑛)) ≠ ∅) |
67 | 64, 66 | sylibr 224 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ≠ ∅) |
68 | | fzfid 12772 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ Fin) |
69 | | ffun 6048 |
. . . . . . . . . . . . 13
⊢ (𝑔:ℕ⟶{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → Fun 𝑔) |
70 | 53, 69 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → Fun 𝑔) |
71 | 70 | adantr 481 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → Fun 𝑔) |
72 | | fores 6124 |
. . . . . . . . . . 11
⊢ ((Fun
𝑔 ∧ (1...𝑛) ⊆ dom 𝑔) → (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛))) |
73 | 71, 57, 72 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛))) |
74 | | fofi 8252 |
. . . . . . . . . 10
⊢
(((1...𝑛) ∈ Fin
∧ (𝑔 ↾ (1...𝑛)):(1...𝑛)–onto→(𝑔 “ (1...𝑛))) → (𝑔 “ (1...𝑛)) ∈ Fin) |
75 | 68, 73, 74 | syl2anc 693 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → (𝑔 “ (1...𝑛)) ∈ Fin) |
76 | | fiinopn 20706 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top → (((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin) → ∩ (𝑔
“ (1...𝑛)) ∈
𝐽)) |
77 | 76 | imp 445 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ ((𝑔 “ (1...𝑛)) ⊆ 𝐽 ∧ (𝑔 “ (1...𝑛)) ≠ ∅ ∧ (𝑔 “ (1...𝑛)) ∈ Fin)) → ∩ (𝑔
“ (1...𝑛)) ∈
𝐽) |
78 | 40, 49, 67, 75, 77 | syl13anc 1328 |
. . . . . . . 8
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑛 ∈ ℕ) → ∩ (𝑔
“ (1...𝑛)) ∈
𝐽) |
79 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) |
80 | 78, 79 | fmptd 6385 |
. . . . . . 7
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))):ℕ⟶𝐽) |
81 | | imassrn 5477 |
. . . . . . . . . . . . 13
⊢ (𝑔 “ (1...𝑘)) ⊆ ran 𝑔 |
82 | 43 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ran 𝑔 = {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
83 | 81, 82 | syl5sseq 3653 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}) |
84 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 𝑛 → 𝐴 ∈ 𝑛) |
85 | 84 | rgenw 2924 |
. . . . . . . . . . . . 13
⊢
∀𝑛 ∈
𝑥 (𝐴 ∈ 𝑛 → 𝐴 ∈ 𝑛) |
86 | | eleq2 2690 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑛 → (𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑛)) |
87 | 86 | ralrab 3368 |
. . . . . . . . . . . . 13
⊢
(∀𝑛 ∈
{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝐴 ∈ 𝑛 ↔ ∀𝑛 ∈ 𝑥 (𝐴 ∈ 𝑛 → 𝐴 ∈ 𝑛)) |
88 | 85, 87 | mpbir 221 |
. . . . . . . . . . . 12
⊢
∀𝑛 ∈
{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝐴 ∈ 𝑛 |
89 | | ssralv 3666 |
. . . . . . . . . . . 12
⊢ ((𝑔 “ (1...𝑘)) ⊆ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → (∀𝑛 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝐴 ∈ 𝑛 → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛)) |
90 | 83, 88, 89 | mpisyl 21 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛) |
91 | | elintg 4483 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ∩ (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛)) |
92 | 91 | ad3antlr 767 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ∩ (𝑔 “ (1...𝑘)) ↔ ∀𝑛 ∈ (𝑔 “ (1...𝑘))𝐴 ∈ 𝑛)) |
93 | 90, 92 | mpbird 247 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ∩ (𝑔 “ (1...𝑘))) |
94 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
95 | 78 | ralrimiva 2966 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑛 ∈ ℕ ∩
(𝑔 “ (1...𝑛)) ∈ 𝐽) |
96 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘)) |
97 | 96 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...𝑘))) |
98 | 97 | inteqd 4480 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → ∩ (𝑔 “ (1...𝑛)) = ∩ (𝑔 “ (1...𝑘))) |
99 | 98 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ↔ ∩ (𝑔 “ (1...𝑘)) ∈ 𝐽)) |
100 | 99 | rspccva 3308 |
. . . . . . . . . . . 12
⊢
((∀𝑛 ∈
ℕ ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ∧ 𝑘 ∈ ℕ) → ∩ (𝑔
“ (1...𝑘)) ∈
𝐽) |
101 | 95, 100 | sylan 488 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∩ (𝑔
“ (1...𝑘)) ∈
𝐽) |
102 | 98, 79 | fvmptg 6280 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧ ∩ (𝑔
“ (1...𝑘)) ∈
𝐽) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) = ∩ (𝑔 “ (1...𝑘))) |
103 | 94, 101, 102 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) = ∩ (𝑔 “ (1...𝑘))) |
104 | 93, 103 | eleqtrrd 2704 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)) |
105 | | fzssp1 12384 |
. . . . . . . . . . . 12
⊢
(1...𝑘) ⊆
(1...(𝑘 +
1)) |
106 | | imass2 5501 |
. . . . . . . . . . . 12
⊢
((1...𝑘) ⊆
(1...(𝑘 + 1)) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1)))) |
107 | 105, 106 | mp1i 13 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1)))) |
108 | | intss 4498 |
. . . . . . . . . . 11
⊢ ((𝑔 “ (1...𝑘)) ⊆ (𝑔 “ (1...(𝑘 + 1))) → ∩
(𝑔 “ (1...(𝑘 + 1))) ⊆ ∩ (𝑔
“ (1...𝑘))) |
109 | 107, 108 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∩ (𝑔
“ (1...(𝑘 + 1)))
⊆ ∩ (𝑔 “ (1...𝑘))) |
110 | | peano2nn 11032 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
111 | 110 | adantl 482 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) |
112 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1))) |
113 | 112 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → (𝑔 “ (1...𝑛)) = (𝑔 “ (1...(𝑘 + 1)))) |
114 | 113 | inteqd 4480 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑘 + 1) → ∩
(𝑔 “ (1...𝑛)) = ∩ (𝑔
“ (1...(𝑘 +
1)))) |
115 | 114 | eleq1d 2686 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑘 + 1) → (∩
(𝑔 “ (1...𝑛)) ∈ 𝐽 ↔ ∩ (𝑔 “ (1...(𝑘 + 1))) ∈ 𝐽)) |
116 | 115 | rspccva 3308 |
. . . . . . . . . . . 12
⊢
((∀𝑛 ∈
ℕ ∩ (𝑔 “ (1...𝑛)) ∈ 𝐽 ∧ (𝑘 + 1) ∈ ℕ) → ∩ (𝑔
“ (1...(𝑘 + 1)))
∈ 𝐽) |
117 | 95, 110, 116 | syl2an 494 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ∩ (𝑔
“ (1...(𝑘 + 1)))
∈ 𝐽) |
118 | 114, 79 | fvmptg 6280 |
. . . . . . . . . . 11
⊢ (((𝑘 + 1) ∈ ℕ ∧ ∩ (𝑔
“ (1...(𝑘 + 1)))
∈ 𝐽) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) = ∩ (𝑔 “ (1...(𝑘 + 1)))) |
119 | 111, 117,
118 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) = ∩ (𝑔 “ (1...(𝑘 + 1)))) |
120 | 109, 119,
103 | 3sstr4d 3648 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)) |
121 | 104, 120 | jca 554 |
. . . . . . . 8
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘))) |
122 | 121 | ralrimiva 2966 |
. . . . . . 7
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘))) |
123 | | simprlr 803 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) |
124 | | eleq2 2690 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑦 → (𝐴 ∈ 𝑧 ↔ 𝐴 ∈ 𝑦)) |
125 | | sseq2 3627 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑦 → (𝑤 ⊆ 𝑧 ↔ 𝑤 ⊆ 𝑦)) |
126 | 125 | anbi2d 740 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑦 → ((𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))) |
127 | 126 | rexbidv 3052 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑦 → (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧) ↔ ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))) |
128 | 124, 127 | imbi12d 334 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑦 → ((𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ↔ (𝐴 ∈ 𝑦 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)))) |
129 | 128 | rspccva 3308 |
. . . . . . . . . . 11
⊢
((∀𝑧 ∈
𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))) |
130 | 123, 129 | sylan 488 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦))) |
131 | | eleq2 2690 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑤 → (𝐴 ∈ 𝑎 ↔ 𝐴 ∈ 𝑤)) |
132 | 131 | rexrab 3370 |
. . . . . . . . . . 11
⊢
(∃𝑤 ∈
{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 ↔ ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦)) |
133 | 43 | rexeqdv 3145 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤 ⊆ 𝑦 ↔ ∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦)) |
134 | | fofn 6117 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → 𝑔 Fn ℕ) |
135 | 134 | ad2antll 765 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → 𝑔 Fn ℕ) |
136 | | sseq1 3626 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = (𝑔‘𝑘) → (𝑤 ⊆ 𝑦 ↔ (𝑔‘𝑘) ⊆ 𝑦)) |
137 | 136 | rexrn 6361 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 Fn ℕ → (∃𝑤 ∈ ran 𝑔 𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦)) |
138 | 135, 137 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑤 ∈ ran 𝑔 𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦)) |
139 | 133, 138 | bitr3d 270 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦)) |
140 | 139 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦)) |
141 | | elfz1end 12371 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (1...𝑘)) |
142 | 70 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → Fun 𝑔) |
143 | | elfznn 12370 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ) |
144 | 143 | ssriv 3607 |
. . . . . . . . . . . . . . . . . 18
⊢
(1...𝑘) ⊆
ℕ |
145 | 55 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → dom 𝑔 = ℕ) |
146 | 144, 145 | syl5sseqr 3654 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (1...𝑘) ⊆ dom 𝑔) |
147 | | funfvima2 6493 |
. . . . . . . . . . . . . . . . 17
⊢ ((Fun
𝑔 ∧ (1...𝑘) ⊆ dom 𝑔) → (𝑘 ∈ (1...𝑘) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘)))) |
148 | 142, 146,
147 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝑘 ∈ (1...𝑘) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘)))) |
149 | 148 | imp 445 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
1st𝜔 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) ∧ 𝑘 ∈ (1...𝑘)) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘))) |
150 | 141, 149 | sylan2b 492 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
1st𝜔 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) ∧ 𝑘 ∈ ℕ) → (𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘))) |
151 | | intss1 4492 |
. . . . . . . . . . . . . 14
⊢ ((𝑔‘𝑘) ∈ (𝑔 “ (1...𝑘)) → ∩ (𝑔 “ (1...𝑘)) ⊆ (𝑔‘𝑘)) |
152 | | sstr2 3610 |
. . . . . . . . . . . . . 14
⊢ (∩ (𝑔
“ (1...𝑘)) ⊆
(𝑔‘𝑘) → ((𝑔‘𝑘) ⊆ 𝑦 → ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
153 | 150, 151,
152 | 3syl 18 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
1st𝜔 ∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) ∧ 𝑘 ∈ ℕ) → ((𝑔‘𝑘) ⊆ 𝑦 → ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
154 | 153 | reximdva 3017 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑘 ∈ ℕ (𝑔‘𝑘) ⊆ 𝑦 → ∃𝑘 ∈ ℕ ∩
(𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
155 | 140, 154 | sylbid 230 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑤 ∈ {𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎}𝑤 ⊆ 𝑦 → ∃𝑘 ∈ ℕ ∩
(𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
156 | 132, 155 | syl5bir 233 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑦) → ∃𝑘 ∈ ℕ ∩
(𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
157 | 130, 156 | syld 47 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ∩
(𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
158 | 103 | sseq1d 3632 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑘 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∩ (𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
159 | 158 | rexbidva 3049 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → (∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ∩
(𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
160 | 159 | adantr 481 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ∩
(𝑔 “ (1...𝑘)) ⊆ 𝑦)) |
161 | 157, 160 | sylibrd 249 |
. . . . . . . 8
⊢ ((((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) ∧ 𝑦 ∈ 𝐽) → (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦)) |
162 | 161 | ralrimiva 2966 |
. . . . . . 7
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦)) |
163 | | nnex 11026 |
. . . . . . . . 9
⊢ ℕ
∈ V |
164 | 163 | mptex 6486 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) ∈
V |
165 | | feq1 6026 |
. . . . . . . . 9
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(𝑓:ℕ⟶𝐽 ↔ (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))):ℕ⟶𝐽)) |
166 | | fveq1 6190 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(𝑓‘𝑘) = ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)) |
167 | 166 | eleq2d 2687 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(𝐴 ∈ (𝑓‘𝑘) ↔ 𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘))) |
168 | | fveq1 6190 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(𝑓‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1))) |
169 | 168, 166 | sseq12d 3634 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
((𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘) ↔ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘))) |
170 | 167, 169 | anbi12d 747 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
((𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ↔ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)))) |
171 | 170 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(∀𝑘 ∈ ℕ
(𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ↔ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)))) |
172 | 166 | sseq1d 3632 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
((𝑓‘𝑘) ⊆ 𝑦 ↔ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦)) |
173 | 172 | rexbidv 3052 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(∃𝑘 ∈ ℕ
(𝑓‘𝑘) ⊆ 𝑦 ↔ ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦)) |
174 | 173 | imbi2d 330 |
. . . . . . . . . 10
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
((𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦) ↔ (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦))) |
175 | 174 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
(∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦) ↔ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦))) |
176 | 165, 171,
175 | 3anbi123d 1399 |
. . . . . . . 8
⊢ (𝑓 = (𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))) →
((𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)) ↔ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))):ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦)))) |
177 | 164, 176 | spcev 3300 |
. . . . . . 7
⊢ (((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛))):ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ∧ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘(𝑘 + 1)) ⊆ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ ((𝑛 ∈ ℕ ↦ ∩ (𝑔
“ (1...𝑛)))‘𝑘) ⊆ 𝑦)) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))) |
178 | 80, 122, 162, 177 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ ((𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))) ∧ 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎})) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))) |
179 | 178 | expr 643 |
. . . . 5
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧)))) → (𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))) |
180 | 179 | adantrrl 760 |
. . . 4
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → (𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))) |
181 | 180 | exlimdv 1861 |
. . 3
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → (∃𝑔 𝑔:ℕ–onto→{𝑎 ∈ 𝑥 ∣ 𝐴 ∈ 𝑎} → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦)))) |
182 | 39, 181 | mpd 15 |
. 2
⊢ (((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) ∧ (𝑥 ∈ 𝒫 𝐽 ∧ (𝑥 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → ∃𝑤 ∈ 𝑥 (𝐴 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑧))))) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))) |
183 | 2, 182 | rexlimddv 3035 |
1
⊢ ((𝐽 ∈ 1st𝜔
∧ 𝐴 ∈ 𝑋) → ∃𝑓(𝑓:ℕ⟶𝐽 ∧ ∀𝑘 ∈ ℕ (𝐴 ∈ (𝑓‘𝑘) ∧ (𝑓‘(𝑘 + 1)) ⊆ (𝑓‘𝑘)) ∧ ∀𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 → ∃𝑘 ∈ ℕ (𝑓‘𝑘) ⊆ 𝑦))) |