Step | Hyp | Ref
| Expression |
1 | | oveq2 7158 |
. . . 4
⊢ (𝑧 = ∅ → (𝐴 +o 𝑧) = (𝐴 +o ∅)) |
2 | | mpteq1 5146 |
. . . . . . . 8
⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥))) |
3 | | mpt0 6484 |
. . . . . . . 8
⊢ (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)) = ∅ |
4 | 2, 3 | syl6eq 2872 |
. . . . . . 7
⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ∅) |
5 | 4 | rneqd 5802 |
. . . . . 6
⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran ∅) |
6 | | rn0 5790 |
. . . . . 6
⊢ ran
∅ = ∅ |
7 | 5, 6 | syl6eq 2872 |
. . . . 5
⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ∅) |
8 | 7 | uneq2d 4138 |
. . . 4
⊢ (𝑧 = ∅ → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ∅)) |
9 | 1, 8 | eqeq12d 2837 |
. . 3
⊢ (𝑧 = ∅ → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o ∅) = (𝐴 ∪ ∅))) |
10 | | oveq2 7158 |
. . . 4
⊢ (𝑧 = 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o 𝑤)) |
11 | | mpteq1 5146 |
. . . . . 6
⊢ (𝑧 = 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) |
12 | 11 | rneqd 5802 |
. . . . 5
⊢ (𝑧 = 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) |
13 | 12 | uneq2d 4138 |
. . . 4
⊢ (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) |
14 | 10, 13 | eqeq12d 2837 |
. . 3
⊢ (𝑧 = 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))))) |
15 | | oveq2 7158 |
. . . 4
⊢ (𝑧 = suc 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑤)) |
16 | | mpteq1 5146 |
. . . . . 6
⊢ (𝑧 = suc 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) |
17 | 16 | rneqd 5802 |
. . . . 5
⊢ (𝑧 = suc 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) |
18 | 17 | uneq2d 4138 |
. . . 4
⊢ (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))) |
19 | 15, 18 | eqeq12d 2837 |
. . 3
⊢ (𝑧 = suc 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))) |
20 | | oveq2 7158 |
. . . 4
⊢ (𝑧 = 𝐵 → (𝐴 +o 𝑧) = (𝐴 +o 𝐵)) |
21 | | mpteq1 5146 |
. . . . . 6
⊢ (𝑧 = 𝐵 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) |
22 | 21 | rneqd 5802 |
. . . . 5
⊢ (𝑧 = 𝐵 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) |
23 | 22 | uneq2d 4138 |
. . . 4
⊢ (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))) |
24 | 20, 23 | eqeq12d 2837 |
. . 3
⊢ (𝑧 = 𝐵 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))) |
25 | | oa0 8135 |
. . . 4
⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴) |
26 | | un0 4343 |
. . . 4
⊢ (𝐴 ∪ ∅) = 𝐴 |
27 | 25, 26 | syl6eqr 2874 |
. . 3
⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 ∪
∅)) |
28 | | uneq1 4131 |
. . . . . 6
⊢ ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)})) |
29 | | unass 4141 |
. . . . . . 7
⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})) |
30 | | rexun 4165 |
. . . . . . . . . . 11
⊢
(∃𝑥 ∈
(𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥))) |
31 | | df-suc 6191 |
. . . . . . . . . . . 12
⊢ suc 𝑤 = (𝑤 ∪ {𝑤}) |
32 | 31 | rexeqi 3414 |
. . . . . . . . . . 11
⊢
(∃𝑥 ∈ suc
𝑤𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥)) |
33 | | eqid 2821 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) |
34 | 33 | elrnmpt 5822 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥))) |
35 | 34 | elv 3499 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥)) |
36 | | velsn 4576 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ 𝑦 = (𝐴 +o 𝑤)) |
37 | | vex 3497 |
. . . . . . . . . . . . . 14
⊢ 𝑤 ∈ V |
38 | | oveq2 7158 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤)) |
39 | 38 | eqeq2d 2832 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑤 → (𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤))) |
40 | 37, 39 | rexsn 4613 |
. . . . . . . . . . . . 13
⊢
(∃𝑥 ∈
{𝑤}𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤)) |
41 | 36, 40 | bitr4i 280 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)) |
42 | 35, 41 | orbi12i 911 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥))) |
43 | 30, 32, 42 | 3bitr4i 305 |
. . . . . . . . . 10
⊢
(∃𝑥 ∈ suc
𝑤𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)})) |
44 | | eqid 2821 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) |
45 | | ovex 7183 |
. . . . . . . . . . 11
⊢ (𝐴 +o 𝑥) ∈ V |
46 | 44, 45 | elrnmpti 5826 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥)) |
47 | | elun 4124 |
. . . . . . . . . 10
⊢ (𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)})) |
48 | 43, 46, 47 | 3bitr4i 305 |
. . . . . . . . 9
⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})) |
49 | 48 | eqriv 2818 |
. . . . . . . 8
⊢ ran
(𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}) |
50 | 49 | uneq2i 4135 |
. . . . . . 7
⊢ (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})) |
51 | 29, 50 | eqtr4i 2847 |
. . . . . 6
⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) |
52 | 28, 51 | syl6eq 2872 |
. . . . 5
⊢ ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))) |
53 | | oasuc 8143 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = suc (𝐴 +o 𝑤)) |
54 | | df-suc 6191 |
. . . . . . 7
⊢ suc
(𝐴 +o 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) |
55 | 53, 54 | syl6eq 2872 |
. . . . . 6
⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)})) |
56 | 55 | eqeq1d 2823 |
. . . . 5
⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) ↔ ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))) |
57 | 52, 56 | syl5ibr 248 |
. . . 4
⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))) |
58 | 57 | expcom 416 |
. . 3
⊢ (𝑤 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))) |
59 | | vex 3497 |
. . . . . . . 8
⊢ 𝑧 ∈ V |
60 | | oalim 8151 |
. . . . . . . 8
⊢ ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤)) |
61 | 59, 60 | mpanr1 701 |
. . . . . . 7
⊢ ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤)) |
62 | 61 | ancoms 461 |
. . . . . 6
⊢ ((Lim
𝑧 ∧ 𝐴 ∈ On) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤)) |
63 | 62 | adantr 483 |
. . . . 5
⊢ (((Lim
𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤)) |
64 | | iuneq2 4930 |
. . . . . 6
⊢
(∀𝑤 ∈
𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) |
65 | 64 | adantl 484 |
. . . . 5
⊢ (((Lim
𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) |
66 | | iunun 5007 |
. . . . . . 7
⊢ ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (∪
𝑤 ∈ 𝑧 𝐴 ∪ ∪
𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) |
67 | | 0ellim 6247 |
. . . . . . . . 9
⊢ (Lim
𝑧 → ∅ ∈
𝑧) |
68 | | ne0i 4299 |
. . . . . . . . 9
⊢ (∅
∈ 𝑧 → 𝑧 ≠ ∅) |
69 | | iunconst 4920 |
. . . . . . . . 9
⊢ (𝑧 ≠ ∅ → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴) |
70 | 67, 68, 69 | 3syl 18 |
. . . . . . . 8
⊢ (Lim
𝑧 → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴) |
71 | | limuni 6245 |
. . . . . . . . . . . 12
⊢ (Lim
𝑧 → 𝑧 = ∪ 𝑧) |
72 | 71 | rexeqdv 3416 |
. . . . . . . . . . 11
⊢ (Lim
𝑧 → (∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥))) |
73 | | df-rex 3144 |
. . . . . . . . . . . . . 14
⊢
(∃𝑥 ∈
𝑤 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
74 | 35, 73 | bitri 277 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
75 | 74 | rexbii 3247 |
. . . . . . . . . . . 12
⊢
(∃𝑤 ∈
𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
76 | | eluni2 4835 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ∪ 𝑧
↔ ∃𝑤 ∈
𝑧 𝑥 ∈ 𝑤) |
77 | 76 | anbi1i 625 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ∪ 𝑧
∧ 𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
78 | | r19.41v 3347 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑤 ∈
𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
79 | 77, 78 | bitr4i 280 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ∪ 𝑧
∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
80 | 79 | exbii 1844 |
. . . . . . . . . . . . 13
⊢
(∃𝑥(𝑥 ∈ ∪ 𝑧
∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
81 | | df-rex 3144 |
. . . . . . . . . . . . 13
⊢
(∃𝑥 ∈
∪ 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥))) |
82 | | rexcom4 3249 |
. . . . . . . . . . . . 13
⊢
(∃𝑤 ∈
𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
83 | 80, 81, 82 | 3bitr4i 305 |
. . . . . . . . . . . 12
⊢
(∃𝑥 ∈
∪ 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥))) |
84 | 75, 83 | bitr4i 280 |
. . . . . . . . . . 11
⊢
(∃𝑤 ∈
𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥)) |
85 | 72, 84 | syl6rbbr 292 |
. . . . . . . . . 10
⊢ (Lim
𝑧 → (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥))) |
86 | | eliun 4915 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) |
87 | | eqid 2821 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) |
88 | 87, 45 | elrnmpti 5826 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥)) |
89 | 85, 86, 88 | 3bitr4g 316 |
. . . . . . . . 9
⊢ (Lim
𝑧 → (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))) |
90 | 89 | eqrdv 2819 |
. . . . . . . 8
⊢ (Lim
𝑧 → ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) |
91 | 70, 90 | uneq12d 4139 |
. . . . . . 7
⊢ (Lim
𝑧 → (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪
𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))) |
92 | 66, 91 | syl5eq 2868 |
. . . . . 6
⊢ (Lim
𝑧 → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))) |
93 | 92 | ad2antrr 724 |
. . . . 5
⊢ (((Lim
𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))) |
94 | 63, 65, 93 | 3eqtrd 2860 |
. . . 4
⊢ (((Lim
𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))) |
95 | 94 | exp31 422 |
. . 3
⊢ (Lim
𝑧 → (𝐴 ∈ On → (∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))))) |
96 | 9, 14, 19, 24, 27, 58, 95 | tfinds3 7573 |
. 2
⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))) |
97 | 96 | impcom 410 |
1
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))) |