Step |
Hyp |
Ref |
Expression |
1 |
|
nn0suc |
⊢ ( 𝑖 ∈ ω → ( 𝑖 = ∅ ∨ ∃ 𝑗 ∈ ω 𝑖 = suc 𝑗 ) ) |
2 |
|
fr0g |
⊢ ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) = Pred ( 𝑅 , 𝐴 , 𝑋 ) ) |
3 |
|
predss |
⊢ Pred ( 𝑅 , 𝐴 , 𝑋 ) ⊆ 𝐴 |
4 |
2 3
|
eqsstrdi |
⊢ ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ⊆ 𝐴 ) |
5 |
|
fveq2 |
⊢ ( 𝑖 = ∅ → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ) |
6 |
5
|
sseq1d |
⊢ ( 𝑖 = ∅ → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ ∅ ) ⊆ 𝐴 ) ) |
7 |
4 6
|
syl5ibr |
⊢ ( 𝑖 = ∅ → ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑋 ) |
9 |
|
nfcv |
⊢ Ⅎ 𝑎 𝑗 |
10 |
|
nfmpt1 |
⊢ Ⅎ 𝑎 ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) |
11 |
10 8
|
nfrdg |
⊢ Ⅎ 𝑎 rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) |
12 |
|
nfcv |
⊢ Ⅎ 𝑎 ω |
13 |
11 12
|
nfres |
⊢ Ⅎ 𝑎 ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) |
14 |
13 9
|
nffv |
⊢ Ⅎ 𝑎 ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) |
15 |
|
nfcv |
⊢ Ⅎ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) |
16 |
14 15
|
nfiun |
⊢ Ⅎ 𝑎 ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) |
17 |
|
predeq3 |
⊢ ( 𝑦 = 𝑒 → Pred ( 𝑅 , 𝐴 , 𝑦 ) = Pred ( 𝑅 , 𝐴 , 𝑒 ) ) |
18 |
17
|
cbviunv |
⊢ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) = ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) |
19 |
18
|
mpteq2i |
⊢ ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) ) |
20 |
|
rdgeq1 |
⊢ ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) = ( 𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) ) → rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) = rec ( ( 𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ) |
21 |
|
reseq1 |
⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) = rec ( ( 𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) → ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) = ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ) |
22 |
19 20 21
|
mp2b |
⊢ ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) = ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) |
23 |
|
iuneq1 |
⊢ ( 𝑎 = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) → ∪ 𝑒 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑒 ) = ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ) |
24 |
8 9 16 22 23
|
frsucmpt |
⊢ ( ( 𝑗 ∈ ω ∧ ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ∈ V ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) = ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ) |
25 |
|
iunss |
⊢ ( ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ⊆ 𝐴 ↔ ∀ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ⊆ 𝐴 ) |
26 |
|
predss |
⊢ Pred ( 𝑅 , 𝐴 , 𝑒 ) ⊆ 𝐴 |
27 |
26
|
a1i |
⊢ ( 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) → Pred ( 𝑅 , 𝐴 , 𝑒 ) ⊆ 𝐴 ) |
28 |
25 27
|
mprgbir |
⊢ ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ⊆ 𝐴 |
29 |
24 28
|
eqsstrdi |
⊢ ( ( 𝑗 ∈ ω ∧ ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ∈ V ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ⊆ 𝐴 ) |
30 |
8 9 16 22 23
|
frsucmptn |
⊢ ( ¬ ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ∈ V → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) = ∅ ) |
31 |
30
|
adantl |
⊢ ( ( 𝑗 ∈ ω ∧ ¬ ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ∈ V ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) = ∅ ) |
32 |
|
0ss |
⊢ ∅ ⊆ 𝐴 |
33 |
31 32
|
eqsstrdi |
⊢ ( ( 𝑗 ∈ ω ∧ ¬ ∪ 𝑒 ∈ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑗 ) Pred ( 𝑅 , 𝐴 , 𝑒 ) ∈ V ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ⊆ 𝐴 ) |
34 |
29 33
|
pm2.61dan |
⊢ ( 𝑗 ∈ ω → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ⊆ 𝐴 ) |
35 |
34
|
adantr |
⊢ ( ( 𝑗 ∈ ω ∧ 𝑖 = suc 𝑗 ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ⊆ 𝐴 ) |
36 |
|
fveq2 |
⊢ ( 𝑖 = suc 𝑗 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) = ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ) |
37 |
36
|
sseq1d |
⊢ ( 𝑖 = suc 𝑗 → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ⊆ 𝐴 ) ) |
38 |
37
|
adantl |
⊢ ( ( 𝑗 ∈ ω ∧ 𝑖 = suc 𝑗 ) → ( ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ↔ ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ suc 𝑗 ) ⊆ 𝐴 ) ) |
39 |
35 38
|
mpbird |
⊢ ( ( 𝑗 ∈ ω ∧ 𝑖 = suc 𝑗 ) → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) |
40 |
39
|
rexlimiva |
⊢ ( ∃ 𝑗 ∈ ω 𝑖 = suc 𝑗 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) |
41 |
40
|
a1d |
⊢ ( ∃ 𝑗 ∈ ω 𝑖 = suc 𝑗 → ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) ) |
42 |
7 41
|
jaoi |
⊢ ( ( 𝑖 = ∅ ∨ ∃ 𝑗 ∈ ω 𝑖 = suc 𝑗 ) → ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) ) |
43 |
1 42
|
syl |
⊢ ( 𝑖 ∈ ω → ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) ) |
44 |
|
nfvres |
⊢ ( ¬ 𝑖 ∈ ω → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) = ∅ ) |
45 |
44 32
|
eqsstrdi |
⊢ ( ¬ 𝑖 ∈ ω → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) |
46 |
45
|
a1d |
⊢ ( ¬ 𝑖 ∈ ω → ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) ) |
47 |
43 46
|
pm2.61i |
⊢ ( Pred ( 𝑅 , 𝐴 , 𝑋 ) ∈ 𝐵 → ( ( rec ( ( 𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred ( 𝑅 , 𝐴 , 𝑦 ) ) , Pred ( 𝑅 , 𝐴 , 𝑋 ) ) ↾ ω ) ‘ 𝑖 ) ⊆ 𝐴 ) |