| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dftrpred2 |
⊢ TrPred ( 𝑅 , 𝐴 , 𝑋 ) = ∪ 𝑖 ∈ ω ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) |
| 2 |
|
fveq2 |
⊢ ( 𝑗 = ∅ → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ) |
| 3 |
2
|
sseq1d |
⊢ ( 𝑗 = ∅ → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ⊆ 𝐵 ) ) |
| 4 |
3
|
imbi2d |
⊢ ( 𝑗 = ∅ → ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ) ↔ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ⊆ 𝐵 ) ) ) |
| 5 |
|
fveq2 |
⊢ ( 𝑗 = 𝑘 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ) |
| 6 |
5
|
sseq1d |
⊢ ( 𝑗 = 𝑘 → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) |
| 7 |
6
|
imbi2d |
⊢ ( 𝑗 = 𝑘 → ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ) ↔ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) ) |
| 8 |
|
fveq2 |
⊢ ( 𝑗 = suc 𝑘 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) ) |
| 9 |
8
|
sseq1d |
⊢ ( 𝑗 = suc 𝑘 → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) ⊆ 𝐵 ) ) |
| 10 |
9
|
imbi2d |
⊢ ( 𝑗 = suc 𝑘 → ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ) ↔ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) ⊆ 𝐵 ) ) ) |
| 11 |
|
fveq2 |
⊢ ( 𝑗 = 𝑖 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ) |
| 12 |
11
|
sseq1d |
⊢ ( 𝑗 = 𝑖 → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) ) |
| 13 |
12
|
imbi2d |
⊢ ( 𝑗 = 𝑖 → ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) ⊆ 𝐵 ) ↔ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) ) ) |
| 14 |
|
setlikespec |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V ) |
| 15 |
|
fr0g |
⊢ ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) ) |
| 16 |
14 15
|
syl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) ) |
| 17 |
16
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) ) |
| 18 |
|
simprr |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) |
| 19 |
17 18
|
eqsstrd |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ⊆ 𝐵 ) |
| 20 |
|
fvex |
⊢ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ∈ V |
| 21 |
|
trpredlem1 |
⊢ ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ V → ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐴 ) |
| 22 |
14 21
|
syl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐴 ) |
| 23 |
22
|
sseld |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) → 𝑦 ∈ 𝐴 ) ) |
| 24 |
|
setlikespec |
⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) |
| 25 |
24
|
expcom |
⊢ ( 𝑅 Se 𝐴 → ( 𝑦 ∈ 𝐴 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) ) |
| 26 |
25
|
adantl |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑦 ∈ 𝐴 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) ) |
| 27 |
23 26
|
syld |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) → Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) ) |
| 28 |
27
|
ralrimiv |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) |
| 29 |
28
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) → ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) |
| 30 |
|
iunexg |
⊢ ( ( ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ∈ V ∧ ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) → ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) |
| 31 |
20 29 30
|
sylancr |
⊢ ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) → ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) |
| 32 |
|
nfcv |
⊢ Ⅎ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑋 ) |
| 33 |
|
nfcv |
⊢ Ⅎ 𝑎 𝑘 |
| 34 |
|
nfcv |
⊢ Ⅎ 𝑎 ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) |
| 35 |
|
eqid |
⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) = ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) |
| 36 |
|
predeq3 |
⊢ ( 𝑦 = 𝑑 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑑 ) ) |
| 37 |
36
|
cbviunv |
⊢ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) = ∪ 𝑑 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑑 ) |
| 38 |
|
iuneq1 |
⊢ ( 𝑎 = 𝑐 → ∪ 𝑑 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑑 ) = ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) |
| 39 |
37 38
|
eqtrid |
⊢ ( 𝑎 = 𝑐 → ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) = ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) |
| 40 |
39
|
cbvmptv |
⊢ ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) |
| 41 |
|
rdgeq1 |
⊢ ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) → rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) = rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) |
| 42 |
|
reseq1 |
⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) = rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) = ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ) |
| 43 |
40 41 42
|
mp2b |
⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) = ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) |
| 44 |
43
|
fveq1i |
⊢ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) = ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) |
| 45 |
44
|
eqeq2i |
⊢ ( 𝑎 = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ↔ 𝑎 = ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ) |
| 46 |
|
iuneq1 |
⊢ ( 𝑎 = ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) → ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) = ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ) |
| 47 |
45 46
|
sylbi |
⊢ ( 𝑎 = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) → ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) = ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ) |
| 48 |
32 33 34 35 47
|
frsucmpt |
⊢ ( ( 𝑘 ∈ ω ∧ ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ∈ V ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) = ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ) |
| 49 |
31 48
|
sylan2 |
⊢ ( ( 𝑘 ∈ ω ∧ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) = ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ) |
| 50 |
44
|
sseq1i |
⊢ ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ↔ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) |
| 51 |
50
|
anbi2i |
⊢ ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ↔ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) |
| 52 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) |
| 53 |
|
nfra1 |
⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 |
| 54 |
|
nfv |
⊢ Ⅎ 𝑦 Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 |
| 55 |
53 54
|
nfan |
⊢ Ⅎ 𝑦 ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) |
| 56 |
52 55
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) |
| 57 |
|
nfv |
⊢ Ⅎ 𝑦 ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 |
| 58 |
56 57
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) |
| 59 |
|
ssel |
⊢ ( ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 → ( 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) → 𝑦 ∈ 𝐵 ) ) |
| 60 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 → ( 𝑦 ∈ 𝐵 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) ) |
| 61 |
60
|
ad2antrl |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( 𝑦 ∈ 𝐵 → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) ) |
| 62 |
59 61
|
sylan9r |
⊢ ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) → ( 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) → Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) ) |
| 63 |
58 62
|
ralrimi |
⊢ ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) → ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) |
| 64 |
63
|
adantl |
⊢ ( ( 𝑘 ∈ ω ∧ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) → ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) |
| 65 |
51 64
|
sylan2b |
⊢ ( ( 𝑘 ∈ ω ∧ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) → ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) |
| 66 |
|
iunss |
⊢ ( ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ↔ ∀ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) |
| 67 |
65 66
|
sylibr |
⊢ ( ( 𝑘 ∈ ω ∧ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) → ∪ 𝑦 ∈ ( ( rec ( ( 𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred ( 𝑅 , 𝐴 , 𝑑 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ) |
| 68 |
49 67
|
eqsstrd |
⊢ ( ( 𝑘 ∈ ω ∧ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) ∧ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) ⊆ 𝐵 ) |
| 69 |
68
|
exp32 |
⊢ ( 𝑘 ∈ ω → ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) ⊆ 𝐵 ) ) ) |
| 70 |
69
|
a2d |
⊢ ( 𝑘 ∈ ω → ( ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑘 ) ⊆ 𝐵 ) → ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑘 ) ⊆ 𝐵 ) ) ) |
| 71 |
4 7 10 13 19 70
|
finds |
⊢ ( 𝑖 ∈ ω → ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) ) |
| 72 |
71
|
com12 |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ( 𝑖 ∈ ω → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) ) |
| 73 |
72
|
ralrimiv |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ∀ 𝑖 ∈ ω ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) |
| 74 |
|
iunss |
⊢ ( ∪ 𝑖 ∈ ω ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ↔ ∀ 𝑖 ∈ ω ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) |
| 75 |
73 74
|
sylibr |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → ∪ 𝑖 ∈ ω ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐵 ) |
| 76 |
1 75
|
eqsstrid |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( ∀ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝐵 ∧ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) ) → TrPred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐵 ) |