Step | Hyp | Ref
| Expression |
1 | | wunex2.u |
. . . . . . . 8
⊢ 𝑈 = ∪
ran 𝐹 |
2 | 1 | eleq2i 2680 |
. . . . . . 7
⊢ (𝑎 ∈ 𝑈 ↔ 𝑎 ∈ ∪ ran
𝐹) |
3 | | frfnom 7417 |
. . . . . . . . 9
⊢
(rec((𝑧 ∈ V
↦ ((𝑧 ∪ ∪ 𝑧)
∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω) Fn ω |
4 | | wunex2.f |
. . . . . . . . . 10
⊢ 𝐹 = (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω) |
5 | 4 | fneq1i 5899 |
. . . . . . . . 9
⊢ (𝐹 Fn ω ↔ (rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧)
∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω) Fn ω) |
6 | 3, 5 | mpbir 220 |
. . . . . . . 8
⊢ 𝐹 Fn ω |
7 | | fnunirn 6415 |
. . . . . . . 8
⊢ (𝐹 Fn ω → (𝑎 ∈ ∪ ran 𝐹 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹‘𝑚))) |
8 | 6, 7 | ax-mp 5 |
. . . . . . 7
⊢ (𝑎 ∈ ∪ ran 𝐹 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹‘𝑚)) |
9 | 2, 8 | bitri 263 |
. . . . . 6
⊢ (𝑎 ∈ 𝑈 ↔ ∃𝑚 ∈ ω 𝑎 ∈ (𝐹‘𝑚)) |
10 | | elssuni 4403 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ (𝐹‘𝑚) → 𝑎 ⊆ ∪ (𝐹‘𝑚)) |
11 | 10 | ad2antll 761 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → 𝑎 ⊆ ∪ (𝐹‘𝑚)) |
12 | | ssun2 3739 |
. . . . . . . . . . 11
⊢ ∪ (𝐹‘𝑚) ⊆ ((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) |
13 | | ssun1 3738 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ⊆ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
14 | 12, 13 | sstri 3577 |
. . . . . . . . . 10
⊢ ∪ (𝐹‘𝑚) ⊆ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
15 | 11, 14 | syl6ss 3580 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → 𝑎 ⊆ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})))) |
16 | | simprl 790 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → 𝑚 ∈ ω) |
17 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝐹‘𝑚) ∈ V |
18 | 17 | uniex 6851 |
. . . . . . . . . . . 12
⊢ ∪ (𝐹‘𝑚) ∈ V |
19 | 17, 18 | unex 6854 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∈ V |
20 | | prex 4836 |
. . . . . . . . . . . . 13
⊢
{𝒫 𝑢, ∪ 𝑢}
∈ V |
21 | 17 | mptex 6390 |
. . . . . . . . . . . . . 14
⊢ (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}) ∈ V |
22 | 21 | rnex 6992 |
. . . . . . . . . . . . 13
⊢ ran
(𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}) ∈ V |
23 | 20, 22 | unex 6854 |
. . . . . . . . . . . 12
⊢
({𝒫 𝑢, ∪ 𝑢}
∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) ∈ V |
24 | 17, 23 | iunex 7039 |
. . . . . . . . . . 11
⊢ ∪ 𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) ∈ V |
25 | 19, 24 | unex 6854 |
. . . . . . . . . 10
⊢ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) ∈ V |
26 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧) |
27 | | unieq 4380 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑧 → ∪ 𝑤 = ∪
𝑧) |
28 | 26, 27 | uneq12d 3730 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑧 → (𝑤 ∪ ∪ 𝑤) = (𝑧 ∪ ∪ 𝑧)) |
29 | | pweq 4111 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑥 → 𝒫 𝑢 = 𝒫 𝑥) |
30 | | unieq 4380 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑥 → ∪ 𝑢 = ∪
𝑥) |
31 | 29, 30 | preq12d 4220 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑥 → {𝒫 𝑢, ∪ 𝑢} = {𝒫 𝑥, ∪
𝑥}) |
32 | | preq2 4213 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝑦 → {𝑢, 𝑣} = {𝑢, 𝑦}) |
33 | 32 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑦 ∈ 𝑤 ↦ {𝑢, 𝑦}) |
34 | | preq1 4212 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 𝑥 → {𝑢, 𝑦} = {𝑥, 𝑦}) |
35 | 34 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑥 → (𝑦 ∈ 𝑤 ↦ {𝑢, 𝑦}) = (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})) |
36 | 33, 35 | syl5eq 2656 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑥 → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})) |
37 | 36 | rneqd 5274 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑥 → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})) |
38 | 31, 37 | uneq12d 3730 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑥 → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦}))) |
39 | 38 | cbviunv 4495 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪
𝑥 ∈ 𝑤 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})) |
40 | | mpteq1 4665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑧 → (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦}) = (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})) |
41 | 40 | rneqd 5274 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑧 → ran (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦}) = ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})) |
42 | 41 | uneq2d 3729 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑧 → ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})) = ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))) |
43 | 26, 42 | iuneq12d 4482 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑧 → ∪
𝑥 ∈ 𝑤 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑤 ↦ {𝑥, 𝑦})) = ∪
𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))) |
44 | 39, 43 | syl5eq 2656 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑧 → ∪
𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪
𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦}))) |
45 | 28, 44 | uneq12d 3730 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑧 → ((𝑤 ∪ ∪ 𝑤) ∪ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}))) = ((𝑧 ∪ ∪ 𝑧) ∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))) |
46 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝐹‘𝑚) → 𝑤 = (𝐹‘𝑚)) |
47 | | unieq 4380 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝐹‘𝑚) → ∪ 𝑤 = ∪
(𝐹‘𝑚)) |
48 | 46, 47 | uneq12d 3730 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝐹‘𝑚) → (𝑤 ∪ ∪ 𝑤) = ((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚))) |
49 | | mpteq1 4665 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝐹‘𝑚) → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) |
50 | 49 | rneqd 5274 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = (𝐹‘𝑚) → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) |
51 | 50 | uneq2d 3729 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝐹‘𝑚) → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
52 | 46, 51 | iuneq12d 4482 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝐹‘𝑚) → ∪
𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
53 | 48, 52 | uneq12d 3730 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝐹‘𝑚) → ((𝑤 ∪ ∪ 𝑤) ∪ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}))) = (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})))) |
54 | 4, 45, 53 | frsucmpt2 7422 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ ω ∧ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) ∈ V) → (𝐹‘suc 𝑚) = (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})))) |
55 | 16, 25, 54 | sylancl 693 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → (𝐹‘suc 𝑚) = (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})))) |
56 | 15, 55 | sseqtr4d 3605 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → 𝑎 ⊆ (𝐹‘suc 𝑚)) |
57 | | fvssunirn 6127 |
. . . . . . . . 9
⊢ (𝐹‘suc 𝑚) ⊆ ∪ ran
𝐹 |
58 | 57, 1 | sseqtr4i 3601 |
. . . . . . . 8
⊢ (𝐹‘suc 𝑚) ⊆ 𝑈 |
59 | 56, 58 | syl6ss 3580 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → 𝑎 ⊆ 𝑈) |
60 | 59 | rexlimdvaa 3014 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹‘𝑚) → 𝑎 ⊆ 𝑈)) |
61 | 9, 60 | syl5bi 231 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → (𝑎 ∈ 𝑈 → 𝑎 ⊆ 𝑈)) |
62 | 61 | ralrimiv 2948 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → ∀𝑎 ∈ 𝑈 𝑎 ⊆ 𝑈) |
63 | | dftr3 4684 |
. . . 4
⊢ (Tr 𝑈 ↔ ∀𝑎 ∈ 𝑈 𝑎 ⊆ 𝑈) |
64 | 62, 63 | sylibr 223 |
. . 3
⊢ (𝐴 ∈ 𝑉 → Tr 𝑈) |
65 | | 1on 7454 |
. . . . . . . 8
⊢
1𝑜 ∈ On |
66 | | unexg 6857 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 1𝑜 ∈ On)
→ (𝐴 ∪
1𝑜) ∈ V) |
67 | 65, 66 | mpan2 703 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1𝑜) ∈
V) |
68 | 4 | fveq1i 6104 |
. . . . . . . 8
⊢ (𝐹‘∅) = ((rec((𝑧 ∈ V ↦ ((𝑧 ∪ ∪ 𝑧)
∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω)‘∅) |
69 | | fr0g 7418 |
. . . . . . . 8
⊢ ((𝐴 ∪ 1𝑜)
∈ V → ((rec((𝑧
∈ V ↦ ((𝑧 ∪
∪ 𝑧) ∪ ∪
𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω)‘∅) = (𝐴 ∪
1𝑜)) |
70 | 68, 69 | syl5eq 2656 |
. . . . . . 7
⊢ ((𝐴 ∪ 1𝑜)
∈ V → (𝐹‘∅) = (𝐴 ∪
1𝑜)) |
71 | 67, 70 | syl 17 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → (𝐹‘∅) = (𝐴 ∪
1𝑜)) |
72 | | fvssunirn 6127 |
. . . . . . 7
⊢ (𝐹‘∅) ⊆ ∪ ran 𝐹 |
73 | 72, 1 | sseqtr4i 3601 |
. . . . . 6
⊢ (𝐹‘∅) ⊆ 𝑈 |
74 | 71, 73 | syl6eqssr 3619 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∪ 1𝑜) ⊆ 𝑈) |
75 | 74 | unssbd 3753 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → 1𝑜 ⊆ 𝑈) |
76 | | 1n0 7462 |
. . . 4
⊢
1𝑜 ≠ ∅ |
77 | | ssn0 3928 |
. . . 4
⊢
((1𝑜 ⊆ 𝑈 ∧ 1𝑜 ≠ ∅)
→ 𝑈 ≠
∅) |
78 | 75, 76, 77 | sylancl 693 |
. . 3
⊢ (𝐴 ∈ 𝑉 → 𝑈 ≠ ∅) |
79 | | pweq 4111 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑎 → 𝒫 𝑢 = 𝒫 𝑎) |
80 | | unieq 4380 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑎 → ∪ 𝑢 = ∪
𝑎) |
81 | 79, 80 | preq12d 4220 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑎 → {𝒫 𝑢, ∪ 𝑢} = {𝒫 𝑎, ∪
𝑎}) |
82 | | preq1 4212 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑎 → {𝑢, 𝑣} = {𝑎, 𝑣}) |
83 | 82 | mpteq2dv 4673 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑎 → (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑎, 𝑣})) |
84 | 83 | rneqd 5274 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑎, 𝑣})) |
85 | 81, 84 | uneq12d 3730 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 𝑎 → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑎, 𝑣}))) |
86 | 85 | ssiun2s 4500 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ (𝐹‘𝑚) → ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑎, 𝑣})) ⊆ ∪ 𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
87 | 86 | ad2antll 761 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑎, 𝑣})) ⊆ ∪ 𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
88 | | ssun2 3739 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) ⊆ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
89 | 88, 55 | syl5sseqr 3617 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → ∪ 𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc 𝑚)) |
90 | 89, 58 | syl6ss 3580 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → ∪ 𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})) ⊆ 𝑈) |
91 | 87, 90 | sstrd 3578 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑎, 𝑣})) ⊆ 𝑈) |
92 | 91 | unssad 3752 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → {𝒫 𝑎, ∪ 𝑎} ⊆ 𝑈) |
93 | | vpwex 4775 |
. . . . . . . . . 10
⊢ 𝒫
𝑎 ∈ V |
94 | | vuniex 6852 |
. . . . . . . . . 10
⊢ ∪ 𝑎
∈ V |
95 | 93, 94 | prss 4291 |
. . . . . . . . 9
⊢
((𝒫 𝑎 ∈
𝑈 ∧ ∪ 𝑎
∈ 𝑈) ↔ {𝒫
𝑎, ∪ 𝑎}
⊆ 𝑈) |
96 | 92, 95 | sylibr 223 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → (𝒫 𝑎 ∈ 𝑈 ∧ ∪ 𝑎 ∈ 𝑈)) |
97 | 96 | simprd 478 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → ∪ 𝑎 ∈ 𝑈) |
98 | 96 | simpld 474 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → 𝒫 𝑎 ∈ 𝑈) |
99 | 1 | eleq2i 2680 |
. . . . . . . . . 10
⊢ (𝑏 ∈ 𝑈 ↔ 𝑏 ∈ ∪ ran
𝐹) |
100 | | fnunirn 6415 |
. . . . . . . . . . 11
⊢ (𝐹 Fn ω → (𝑏 ∈ ∪ ran 𝐹 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹‘𝑛))) |
101 | 6, 100 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝑏 ∈ ∪ ran 𝐹 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹‘𝑛)) |
102 | 99, 101 | bitri 263 |
. . . . . . . . 9
⊢ (𝑏 ∈ 𝑈 ↔ ∃𝑛 ∈ ω 𝑏 ∈ (𝐹‘𝑛)) |
103 | | simplrl 796 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → 𝑚 ∈ ω) |
104 | | simprl 790 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → 𝑛 ∈ ω) |
105 | | ordom 6966 |
. . . . . . . . . . . . . . . . . 18
⊢ Ord
ω |
106 | | ordunel 6919 |
. . . . . . . . . . . . . . . . . 18
⊢ ((Ord
ω ∧ 𝑚 ∈
ω ∧ 𝑛 ∈
ω) → (𝑚 ∪
𝑛) ∈
ω) |
107 | 105, 106 | mp3an1 1403 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (𝑚 ∪ 𝑛) ∈ ω) |
108 | 103, 104,
107 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → (𝑚 ∪ 𝑛) ∈ ω) |
109 | | ssun1 3738 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑚 ⊆ (𝑚 ∪ 𝑛) |
110 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) |
111 | 110 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑚 → ((𝐹‘𝑚) ⊆ (𝐹‘𝑘) ↔ (𝐹‘𝑚) ⊆ (𝐹‘𝑚))) |
112 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑖 → (𝐹‘𝑘) = (𝐹‘𝑖)) |
113 | 112 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑖 → ((𝐹‘𝑚) ⊆ (𝐹‘𝑘) ↔ (𝐹‘𝑚) ⊆ (𝐹‘𝑖))) |
114 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = suc 𝑖 → (𝐹‘𝑘) = (𝐹‘suc 𝑖)) |
115 | 114 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = suc 𝑖 → ((𝐹‘𝑚) ⊆ (𝐹‘𝑘) ↔ (𝐹‘𝑚) ⊆ (𝐹‘suc 𝑖))) |
116 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = (𝑚 ∪ 𝑛) → (𝐹‘𝑘) = (𝐹‘(𝑚 ∪ 𝑛))) |
117 | 116 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = (𝑚 ∪ 𝑛) → ((𝐹‘𝑚) ⊆ (𝐹‘𝑘) ↔ (𝐹‘𝑚) ⊆ (𝐹‘(𝑚 ∪ 𝑛)))) |
118 | | ssid 3587 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹‘𝑚) ⊆ (𝐹‘𝑚) |
119 | 118 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ω → (𝐹‘𝑚) ⊆ (𝐹‘𝑚)) |
120 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 = 𝑖 → (𝐹‘𝑚) = (𝐹‘𝑖)) |
121 | | suceq 5707 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 = 𝑖 → suc 𝑚 = suc 𝑖) |
122 | 121 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 = 𝑖 → (𝐹‘suc 𝑚) = (𝐹‘suc 𝑖)) |
123 | 120, 122 | sseq12d 3597 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 = 𝑖 → ((𝐹‘𝑚) ⊆ (𝐹‘suc 𝑚) ↔ (𝐹‘𝑖) ⊆ (𝐹‘suc 𝑖))) |
124 | | ssun1 3738 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹‘𝑚) ⊆ ((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) |
125 | 124, 13 | sstri 3577 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹‘𝑚) ⊆ (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣}))) |
126 | 25, 54 | mpan2 703 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ ω → (𝐹‘suc 𝑚) = (((𝐹‘𝑚) ∪ ∪ (𝐹‘𝑚)) ∪ ∪
𝑢 ∈ (𝐹‘𝑚)({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘𝑚) ↦ {𝑢, 𝑣})))) |
127 | 125, 126 | syl5sseqr 3617 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ ω → (𝐹‘𝑚) ⊆ (𝐹‘suc 𝑚)) |
128 | 123, 127 | vtoclga 3245 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ ω → (𝐹‘𝑖) ⊆ (𝐹‘suc 𝑖)) |
129 | 128 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ 𝑖) → (𝐹‘𝑖) ⊆ (𝐹‘suc 𝑖)) |
130 | | sstr2 3575 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹‘𝑚) ⊆ (𝐹‘𝑖) → ((𝐹‘𝑖) ⊆ (𝐹‘suc 𝑖) → (𝐹‘𝑚) ⊆ (𝐹‘suc 𝑖))) |
131 | 129, 130 | syl5com 31 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ 𝑖) → ((𝐹‘𝑚) ⊆ (𝐹‘𝑖) → (𝐹‘𝑚) ⊆ (𝐹‘suc 𝑖))) |
132 | 111, 113,
115, 117, 119, 131 | findsg 6985 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ (𝑚 ∪ 𝑛)) → (𝐹‘𝑚) ⊆ (𝐹‘(𝑚 ∪ 𝑛))) |
133 | 109, 132 | mpan2 703 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑚 ∪ 𝑛) ∈ ω ∧ 𝑚 ∈ ω) → (𝐹‘𝑚) ⊆ (𝐹‘(𝑚 ∪ 𝑛))) |
134 | 108, 103,
133 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → (𝐹‘𝑚) ⊆ (𝐹‘(𝑚 ∪ 𝑛))) |
135 | | simplrr 797 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → 𝑎 ∈ (𝐹‘𝑚)) |
136 | 134, 135 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → 𝑎 ∈ (𝐹‘(𝑚 ∪ 𝑛))) |
137 | 82 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑎 → (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) |
138 | 137 | rneqd 5274 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑎 → ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) |
139 | 81, 138 | uneq12d 3730 |
. . . . . . . . . . . . . . 15
⊢ (𝑢 = 𝑎 → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) = ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣}))) |
140 | 139 | ssiun2s 4500 |
. . . . . . . . . . . . . 14
⊢ (𝑎 ∈ (𝐹‘(𝑚 ∪ 𝑛)) → ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) ⊆ ∪ 𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) |
141 | 136, 140 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) ⊆ ∪ 𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) |
142 | | ssun2 3739 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) ⊆ (((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∪ ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) |
143 | | fvex 6113 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹‘(𝑚 ∪ 𝑛)) ∈ V |
144 | 143 | uniex 6851 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ (𝐹‘(𝑚 ∪ 𝑛)) ∈ V |
145 | 143, 144 | unex 6854 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∈ V |
146 | 143 | mptex 6390 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}) ∈ V |
147 | 146 | rnex 6992 |
. . . . . . . . . . . . . . . . . . 19
⊢ ran
(𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}) ∈ V |
148 | 20, 147 | unex 6854 |
. . . . . . . . . . . . . . . . . 18
⊢
({𝒫 𝑢, ∪ 𝑢}
∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) ∈ V |
149 | 143, 148 | iunex 7039 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) ∈ V |
150 | 145, 149 | unex 6854 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∪ ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) ∈ V |
151 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → 𝑤 = (𝐹‘(𝑚 ∪ 𝑛))) |
152 | | unieq 4380 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → ∪ 𝑤 = ∪
(𝐹‘(𝑚 ∪ 𝑛))) |
153 | 151, 152 | uneq12d 3730 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → (𝑤 ∪ ∪ 𝑤) = ((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛)))) |
154 | | mpteq1 4665 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) |
155 | 154 | rneqd 5274 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}) = ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) |
156 | 155 | uneq2d 3729 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) |
157 | 151, 156 | iuneq12d 4482 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → ∪
𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣})) = ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) |
158 | 153, 157 | uneq12d 3730 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = (𝐹‘(𝑚 ∪ 𝑛)) → ((𝑤 ∪ ∪ 𝑤) ∪ ∪ 𝑢 ∈ 𝑤 ({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ 𝑤 ↦ {𝑢, 𝑣}))) = (((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∪ ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})))) |
159 | 4, 45, 158 | frsucmpt2 7422 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑚 ∪ 𝑛) ∈ ω ∧ (((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∪ ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣}))) ∈ V) → (𝐹‘suc (𝑚 ∪ 𝑛)) = (((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∪ ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})))) |
160 | 108, 150,
159 | sylancl 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → (𝐹‘suc (𝑚 ∪ 𝑛)) = (((𝐹‘(𝑚 ∪ 𝑛)) ∪ ∪ (𝐹‘(𝑚 ∪ 𝑛))) ∪ ∪
𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})))) |
161 | 142, 160 | syl5sseqr 3617 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → ∪ 𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) ⊆ (𝐹‘suc (𝑚 ∪ 𝑛))) |
162 | | fvssunirn 6127 |
. . . . . . . . . . . . . . 15
⊢ (𝐹‘suc (𝑚 ∪ 𝑛)) ⊆ ∪ ran
𝐹 |
163 | 162, 1 | sseqtr4i 3601 |
. . . . . . . . . . . . . 14
⊢ (𝐹‘suc (𝑚 ∪ 𝑛)) ⊆ 𝑈 |
164 | 161, 163 | syl6ss 3580 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → ∪ 𝑢 ∈ (𝐹‘(𝑚 ∪ 𝑛))({𝒫 𝑢, ∪ 𝑢} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑢, 𝑣})) ⊆ 𝑈) |
165 | 141, 164 | sstrd 3578 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → ({𝒫 𝑎, ∪ 𝑎} ∪ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) ⊆ 𝑈) |
166 | 165 | unssbd 3753 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣}) ⊆ 𝑈) |
167 | | ssun2 3739 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑛 ⊆ (𝑚 ∪ 𝑛) |
168 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = (𝑚 ∪ 𝑛) → 𝑖 = (𝑚 ∪ 𝑛)) |
169 | 167, 168 | syl5sseqr 3617 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = (𝑚 ∪ 𝑛) → 𝑛 ⊆ 𝑖) |
170 | 169 | biantrud 527 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑚 ∪ 𝑛) → (𝑛 ∈ ω ↔ (𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑖))) |
171 | 170 | bicomd 212 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = (𝑚 ∪ 𝑛) → ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑖) ↔ 𝑛 ∈ ω)) |
172 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = (𝑚 ∪ 𝑛) → (𝐹‘𝑖) = (𝐹‘(𝑚 ∪ 𝑛))) |
173 | 172 | sseq2d 3596 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = (𝑚 ∪ 𝑛) → ((𝐹‘𝑛) ⊆ (𝐹‘𝑖) ↔ (𝐹‘𝑛) ⊆ (𝐹‘(𝑚 ∪ 𝑛)))) |
174 | 171, 173 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = (𝑚 ∪ 𝑛) → (((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑖) → (𝐹‘𝑛) ⊆ (𝐹‘𝑖)) ↔ (𝑛 ∈ ω → (𝐹‘𝑛) ⊆ (𝐹‘(𝑚 ∪ 𝑛))))) |
175 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = 𝑛 → (𝑚 ∈ ω ↔ 𝑛 ∈ ω)) |
176 | 175 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = 𝑛 → ((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ↔ (𝑖 ∈ ω ∧ 𝑛 ∈ ω))) |
177 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = 𝑛 → (𝑚 ⊆ 𝑖 ↔ 𝑛 ⊆ 𝑖)) |
178 | 176, 177 | anbi12d 743 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑛 → (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ 𝑖) ↔ ((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛 ⊆ 𝑖))) |
179 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛)) |
180 | 179 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ (𝐹‘𝑖) ↔ (𝐹‘𝑛) ⊆ (𝐹‘𝑖))) |
181 | 178, 180 | imbi12d 333 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑛 → ((((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ 𝑖) → (𝐹‘𝑚) ⊆ (𝐹‘𝑖)) ↔ (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛 ⊆ 𝑖) → (𝐹‘𝑛) ⊆ (𝐹‘𝑖)))) |
182 | 111, 113,
115, 113, 119, 131 | findsg 6985 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑖 ∈ ω ∧ 𝑚 ∈ ω) ∧ 𝑚 ⊆ 𝑖) → (𝐹‘𝑚) ⊆ (𝐹‘𝑖)) |
183 | 181, 182 | chvarv 2251 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑖 ∈ ω ∧ 𝑛 ∈ ω) ∧ 𝑛 ⊆ 𝑖) → (𝐹‘𝑛) ⊆ (𝐹‘𝑖)) |
184 | 183 | expl 646 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ ω → ((𝑛 ∈ ω ∧ 𝑛 ⊆ 𝑖) → (𝐹‘𝑛) ⊆ (𝐹‘𝑖))) |
185 | 174, 184 | vtoclga 3245 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 ∪ 𝑛) ∈ ω → (𝑛 ∈ ω → (𝐹‘𝑛) ⊆ (𝐹‘(𝑚 ∪ 𝑛)))) |
186 | 108, 104,
185 | sylc 63 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → (𝐹‘𝑛) ⊆ (𝐹‘(𝑚 ∪ 𝑛))) |
187 | | simprr 792 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → 𝑏 ∈ (𝐹‘𝑛)) |
188 | 186, 187 | sseldd 3569 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → 𝑏 ∈ (𝐹‘(𝑚 ∪ 𝑛))) |
189 | | prex 4836 |
. . . . . . . . . . . 12
⊢ {𝑎, 𝑏} ∈ V |
190 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣}) = (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣}) |
191 | | preq2 4213 |
. . . . . . . . . . . . 13
⊢ (𝑣 = 𝑏 → {𝑎, 𝑣} = {𝑎, 𝑏}) |
192 | 190, 191 | elrnmpt1s 5294 |
. . . . . . . . . . . 12
⊢ ((𝑏 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ∧ {𝑎, 𝑏} ∈ V) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) |
193 | 188, 189,
192 | sylancl 693 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → {𝑎, 𝑏} ∈ ran (𝑣 ∈ (𝐹‘(𝑚 ∪ 𝑛)) ↦ {𝑎, 𝑣})) |
194 | 166, 193 | sseldd 3569 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) ∧ (𝑛 ∈ ω ∧ 𝑏 ∈ (𝐹‘𝑛))) → {𝑎, 𝑏} ∈ 𝑈) |
195 | 194 | rexlimdvaa 3014 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → (∃𝑛 ∈ ω 𝑏 ∈ (𝐹‘𝑛) → {𝑎, 𝑏} ∈ 𝑈)) |
196 | 102, 195 | syl5bi 231 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → (𝑏 ∈ 𝑈 → {𝑎, 𝑏} ∈ 𝑈)) |
197 | 196 | ralrimiv 2948 |
. . . . . . 7
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈) |
198 | 97, 98, 197 | 3jca 1235 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ (𝑚 ∈ ω ∧ 𝑎 ∈ (𝐹‘𝑚))) → (∪
𝑎 ∈ 𝑈 ∧ 𝒫 𝑎 ∈ 𝑈 ∧ ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈)) |
199 | 198 | rexlimdvaa 3014 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → (∃𝑚 ∈ ω 𝑎 ∈ (𝐹‘𝑚) → (∪ 𝑎 ∈ 𝑈 ∧ 𝒫 𝑎 ∈ 𝑈 ∧ ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈))) |
200 | 9, 199 | syl5bi 231 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → (𝑎 ∈ 𝑈 → (∪ 𝑎 ∈ 𝑈 ∧ 𝒫 𝑎 ∈ 𝑈 ∧ ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈))) |
201 | 200 | ralrimiv 2948 |
. . 3
⊢ (𝐴 ∈ 𝑉 → ∀𝑎 ∈ 𝑈 (∪ 𝑎 ∈ 𝑈 ∧ 𝒫 𝑎 ∈ 𝑈 ∧ ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈)) |
202 | | rdgfun 7399 |
. . . . . . . . 9
⊢ Fun
rec((𝑧 ∈ V ↦
((𝑧 ∪ ∪ 𝑧)
∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪
1𝑜)) |
203 | | omex 8423 |
. . . . . . . . 9
⊢ ω
∈ V |
204 | | resfunexg 6384 |
. . . . . . . . 9
⊢ ((Fun
rec((𝑧 ∈ V ↦
((𝑧 ∪ ∪ 𝑧)
∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ∧ ω
∈ V) → (rec((𝑧
∈ V ↦ ((𝑧 ∪
∪ 𝑧) ∪ ∪
𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω) ∈ V) |
205 | 202, 203,
204 | mp2an 704 |
. . . . . . . 8
⊢
(rec((𝑧 ∈ V
↦ ((𝑧 ∪ ∪ 𝑧)
∪ ∪ 𝑥 ∈ 𝑧 ({𝒫 𝑥, ∪ 𝑥} ∪ ran (𝑦 ∈ 𝑧 ↦ {𝑥, 𝑦})))), (𝐴 ∪ 1𝑜)) ↾
ω) ∈ V |
206 | 4, 205 | eqeltri 2684 |
. . . . . . 7
⊢ 𝐹 ∈ V |
207 | 206 | rnex 6992 |
. . . . . 6
⊢ ran 𝐹 ∈ V |
208 | 207 | uniex 6851 |
. . . . 5
⊢ ∪ ran 𝐹 ∈ V |
209 | 1, 208 | eqeltri 2684 |
. . . 4
⊢ 𝑈 ∈ V |
210 | | iswun 9405 |
. . . 4
⊢ (𝑈 ∈ V → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑎 ∈ 𝑈 (∪ 𝑎 ∈ 𝑈 ∧ 𝒫 𝑎 ∈ 𝑈 ∧ ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈)))) |
211 | 209, 210 | ax-mp 5 |
. . 3
⊢ (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑎 ∈ 𝑈 (∪ 𝑎 ∈ 𝑈 ∧ 𝒫 𝑎 ∈ 𝑈 ∧ ∀𝑏 ∈ 𝑈 {𝑎, 𝑏} ∈ 𝑈))) |
212 | 64, 78, 201, 211 | syl3anbrc 1239 |
. 2
⊢ (𝐴 ∈ 𝑉 → 𝑈 ∈ WUni) |
213 | 74 | unssad 3752 |
. 2
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ 𝑈) |
214 | 212, 213 | jca 553 |
1
⊢ (𝐴 ∈ 𝑉 → (𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈)) |