Step | Hyp | Ref
| Expression |
1 | | df-finxp 35793 |
. . 3
⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} |
2 | 1 | csbeq2i 3861 |
. 2
⊢
⦋𝐴 /
𝑥⦌(𝑈↑↑𝑁) = ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} |
3 | | sbcan 3789 |
. . . . 5
⊢
([𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ ([𝐴 / 𝑥]𝑁 ∈ ω ∧ [𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))) |
4 | | sbcel1g 4371 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑁 ∈ ω ↔ ⦋𝐴 / 𝑥⦌𝑁 ∈ ω)) |
5 | | sbceq2g 4374 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))) |
6 | | csbfv12 6887 |
. . . . . . . . 9
⊢
⦋𝐴 /
𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁) |
7 | | csbrdgg 35738 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩) = rec(⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩)) |
8 | | csbmpo123 35740 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ⦋𝐴 / 𝑥⦌ω, 𝑧 ∈ ⦋𝐴 / 𝑥⦌V ↦ ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)))) |
9 | | csbconstg 3872 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ω =
ω) |
10 | | csbconstg 3872 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌V = V) |
11 | | csbif 4541 |
. . . . . . . . . . . . . . 15
⊢
⦋𝐴 /
𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ⦋𝐴 / 𝑥⦌∅, ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) |
12 | | sbcan 3789 |
. . . . . . . . . . . . . . . . 17
⊢
([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈) ↔ ([𝐴 / 𝑥]𝑛 = 1o ∧ [𝐴 / 𝑥]𝑧 ∈ 𝑈)) |
13 | | sbcg 3816 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑛 = 1o ↔ 𝑛 = 1o)) |
14 | | sbcel12 4366 |
. . . . . . . . . . . . . . . . . . 19
⊢
([𝐴 / 𝑥]𝑧 ∈ 𝑈 ↔ ⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈) |
15 | | csbconstg 3872 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧) |
16 | 15 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈)) |
17 | 14, 16 | bitrid 282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ 𝑈 ↔ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈)) |
18 | 13, 17 | anbi12d 631 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑛 = 1o ∧ [𝐴 / 𝑥]𝑧 ∈ 𝑈) ↔ (𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈))) |
19 | 12, 18 | bitrid 282 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈) ↔ (𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈))) |
20 | | csbconstg 3872 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌∅ =
∅) |
21 | | csbif 4541 |
. . . . . . . . . . . . . . . . 17
⊢
⦋𝐴 /
𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛,
(1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩) = if([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈), ⦋𝐴 / 𝑥⦌⟨∪ 𝑛,
(1st ‘𝑧)⟩, ⦋𝐴 / 𝑥⦌⟨𝑛, 𝑧⟩) |
22 | | sbcel12 4366 |
. . . . . . . . . . . . . . . . . . 19
⊢
([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ ⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌(V × 𝑈)) |
23 | | csbxp 5729 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
⦋𝐴 /
𝑥⦌(V ×
𝑈) = (⦋𝐴 / 𝑥⦌V × ⦋𝐴 / 𝑥⦌𝑈) |
24 | 10 | xpeq1d 5660 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌V × ⦋𝐴 / 𝑥⦌𝑈) = (V × ⦋𝐴 / 𝑥⦌𝑈)) |
25 | 23, 24 | eqtrid 2789 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(V × 𝑈) = (V × ⦋𝐴 / 𝑥⦌𝑈)) |
26 | 15, 25 | eleq12d 2832 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧 ∈ ⦋𝐴 / 𝑥⦌(V × 𝑈) ↔ 𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈))) |
27 | 22, 26 | bitrid 282 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈) ↔ 𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈))) |
28 | | csbconstg 3872 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨∪ 𝑛,
(1st ‘𝑧)⟩ = ⟨∪
𝑛, (1st
‘𝑧)⟩) |
29 | | csbconstg 3872 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑛, 𝑧⟩ = ⟨𝑛, 𝑧⟩) |
30 | 27, 28, 29 | ifbieq12d 4512 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥]𝑧 ∈ (V × 𝑈), ⦋𝐴 / 𝑥⦌⟨∪ 𝑛,
(1st ‘𝑧)⟩, ⦋𝐴 / 𝑥⦌⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) |
31 | 21, 30 | eqtrid 2789 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩) = if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) |
32 | 19, 20, 31 | ifbieq12d 4512 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ∈ 𝑉 → if([𝐴 / 𝑥](𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ⦋𝐴 / 𝑥⦌∅, ⦋𝐴 / 𝑥⦌if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) |
33 | 11, 32 | eqtrid 2789 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)) = if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) |
34 | 9, 10, 33 | mpoeq123dv 7426 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑉 → (𝑛 ∈ ⦋𝐴 / 𝑥⦌ω, 𝑧 ∈ ⦋𝐴 / 𝑥⦌V ↦ ⦋𝐴 / 𝑥⦌if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)))) |
35 | 8, 34 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩)))) |
36 | | csbopg 4846 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, ⦋𝐴 / 𝑥⦌𝑦⟩) |
37 | | csbconstg 3872 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦) |
38 | 37 | opeq2d 4835 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ 𝑉 → ⟨⦋𝐴 / 𝑥⦌𝑁, ⦋𝐴 / 𝑥⦌𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩) |
39 | 36, 38 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩) |
40 | | rdgeq12 8351 |
. . . . . . . . . . . 12
⊢
((⦋𝐴 /
𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) = (𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))) ∧ ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩ = ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩) → rec(⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)) |
41 | 35, 39, 40 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑉 → rec(⦋𝐴 / 𝑥⦌(𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⦋𝐴 / 𝑥⦌⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)) |
42 | 7, 41 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)) |
43 | 42 | fveq1d 6841 |
. . . . . . . . 9
⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁) = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)) |
44 | 6, 43 | eqtrid 2789 |
. . . . . . . 8
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)) |
45 | 44 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝐴 ∈ 𝑉 → (∅ = ⦋𝐴 / 𝑥⦌(rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))) |
46 | 5, 45 | bitrd 278 |
. . . . . 6
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))) |
47 | 4, 46 | anbi12d 631 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝑁 ∈ ω ∧ [𝐴 / 𝑥]∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)))) |
48 | 3, 47 | bitrid 282 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁)))) |
49 | 48 | abbidv 2806 |
. . 3
⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))}) |
50 | | csbab 4395 |
. . 3
⊢
⦋𝐴 /
𝑥⦌{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ [𝐴 / 𝑥](𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} |
51 | | df-finxp 35793 |
. . 3
⊢
(⦋𝐴 /
𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁) = {𝑦 ∣ (⦋𝐴 / 𝑥⦌𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑈), ∅, if(𝑧 ∈ (V × ⦋𝐴 / 𝑥⦌𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨⦋𝐴 / 𝑥⦌𝑁, 𝑦⟩)‘⦋𝐴 / 𝑥⦌𝑁))} |
52 | 49, 50, 51 | 3eqtr4g 2802 |
. 2
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑧 ∈ V ↦ if((𝑛 = 1o ∧ 𝑧 ∈ 𝑈), ∅, if(𝑧 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑧)⟩, ⟨𝑛, 𝑧⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁)) |
53 | 2, 52 | eqtrid 2789 |
1
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁)) |