Step |
Hyp |
Ref |
Expression |
1 |
|
rdgssun.1 |
⊢ 𝐹 = ( 𝑤 ∈ V ↦ ( 𝑤 ∪ 𝐵 ) ) |
2 |
|
rdgssun.2 |
⊢ 𝐵 ∈ V |
3 |
|
nfsbc1v |
⊢ Ⅎ 𝑥 [ ∅ / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) |
4 |
|
0ex |
⊢ ∅ ∈ V |
5 |
|
rzal |
⊢ ( 𝑥 = ∅ → ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) |
6 |
|
sbceq1a |
⊢ ( 𝑥 = ∅ → ( ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ [ ∅ / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) |
7 |
5 6
|
mpbid |
⊢ ( 𝑥 = ∅ → [ ∅ / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) |
8 |
3 4 7
|
vtoclef |
⊢ [ ∅ / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) |
9 |
|
vex |
⊢ 𝑦 ∈ V |
10 |
9
|
elsuc |
⊢ ( 𝑦 ∈ suc 𝑥 ↔ ( 𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ) ) |
11 |
|
ssun1 |
⊢ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) |
12 |
|
fvex |
⊢ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∈ V |
13 |
2
|
csbex |
⊢ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ∈ V |
14 |
12 13
|
unex |
⊢ ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) ∈ V |
15 |
|
nfcv |
⊢ Ⅎ 𝑤 𝐴 |
16 |
|
nfcv |
⊢ Ⅎ 𝑤 𝑥 |
17 |
|
nfmpt1 |
⊢ Ⅎ 𝑤 ( 𝑤 ∈ V ↦ ( 𝑤 ∪ 𝐵 ) ) |
18 |
1 17
|
nfcxfr |
⊢ Ⅎ 𝑤 𝐹 |
19 |
18 15
|
nfrdg |
⊢ Ⅎ 𝑤 rec ( 𝐹 , 𝐴 ) |
20 |
19 16
|
nffv |
⊢ Ⅎ 𝑤 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) |
21 |
20
|
nfcsb1 |
⊢ Ⅎ 𝑤 ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 |
22 |
20 21
|
nfun |
⊢ Ⅎ 𝑤 ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) |
23 |
|
rdgeq1 |
⊢ ( 𝐹 = ( 𝑤 ∈ V ↦ ( 𝑤 ∪ 𝐵 ) ) → rec ( 𝐹 , 𝐴 ) = rec ( ( 𝑤 ∈ V ↦ ( 𝑤 ∪ 𝐵 ) ) , 𝐴 ) ) |
24 |
1 23
|
ax-mp |
⊢ rec ( 𝐹 , 𝐴 ) = rec ( ( 𝑤 ∈ V ↦ ( 𝑤 ∪ 𝐵 ) ) , 𝐴 ) |
25 |
|
id |
⊢ ( 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) |
26 |
|
csbeq1a |
⊢ ( 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → 𝐵 = ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) |
27 |
25 26
|
uneq12d |
⊢ ( 𝑤 = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( 𝑤 ∪ 𝐵 ) = ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) ) |
28 |
15 16 22 24 27
|
rdgsucmptf |
⊢ ( ( 𝑥 ∈ On ∧ ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) ∈ V ) → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) = ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) ) |
29 |
14 28
|
mpan2 |
⊢ ( 𝑥 ∈ On → ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) = ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ∪ ⦋ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) / 𝑤 ⦌ 𝐵 ) ) |
30 |
11 29
|
sseqtrrid |
⊢ ( 𝑥 ∈ On → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) |
31 |
|
sstr2 |
⊢ ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
32 |
30 31
|
syl5com |
⊢ ( 𝑥 ∈ On → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
33 |
32
|
imim2d |
⊢ ( 𝑥 ∈ On → ( ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) → ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) ) |
34 |
33
|
imp |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
35 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) |
36 |
35
|
sseq1d |
⊢ ( 𝑦 = 𝑥 → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
37 |
30 36
|
syl5ibrcom |
⊢ ( 𝑥 ∈ On → ( 𝑦 = 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( 𝑦 = 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
39 |
34 38
|
jaod |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( ( 𝑦 ∈ 𝑥 ∨ 𝑦 = 𝑥 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
40 |
10 39
|
syl5bi |
⊢ ( ( 𝑥 ∈ On ∧ ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) → ( 𝑦 ∈ suc 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
41 |
40
|
ex |
⊢ ( 𝑥 ∈ On → ( ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) → ( 𝑦 ∈ suc 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) ) |
42 |
41
|
ralimdv2 |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
43 |
|
df-sbc |
⊢ ( [ suc 𝑥 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ suc 𝑥 ∈ { 𝑥 ∣ ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } ) |
44 |
|
vex |
⊢ 𝑥 ∈ V |
45 |
44
|
sucex |
⊢ suc 𝑥 ∈ V |
46 |
|
fveq2 |
⊢ ( 𝑧 = suc 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) |
47 |
46
|
sseq2d |
⊢ ( 𝑧 = suc 𝑥 → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
48 |
47
|
raleqbi1dv |
⊢ ( 𝑧 = suc 𝑥 → ( ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ↔ ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) ) |
49 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) |
50 |
49
|
sseq2d |
⊢ ( 𝑥 = 𝑧 → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) ) |
51 |
50
|
raleqbi1dv |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) ) |
52 |
51
|
cbvabv |
⊢ { 𝑥 ∣ ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } = { 𝑧 ∣ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } |
53 |
45 48 52
|
elab2 |
⊢ ( suc 𝑥 ∈ { 𝑥 ∣ ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } ↔ ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) |
54 |
43 53
|
bitri |
⊢ ( [ suc 𝑥 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ∀ 𝑦 ∈ suc 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ suc 𝑥 ) ) |
55 |
42 54
|
syl6ibr |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → [ suc 𝑥 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) |
56 |
|
ssiun2 |
⊢ ( 𝑦 ∈ 𝑧 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ∪ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ) |
57 |
56
|
adantl |
⊢ ( ( Lim 𝑧 ∧ 𝑦 ∈ 𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ∪ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ) |
58 |
|
vex |
⊢ 𝑧 ∈ V |
59 |
|
rdglim2a |
⊢ ( ( 𝑧 ∈ V ∧ Lim 𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = ∪ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ) |
60 |
58 59
|
mpan |
⊢ ( Lim 𝑧 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = ∪ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ) |
61 |
60
|
adantr |
⊢ ( ( Lim 𝑧 ∧ 𝑦 ∈ 𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) = ∪ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ) |
62 |
57 61
|
sseqtrrd |
⊢ ( ( Lim 𝑧 ∧ 𝑦 ∈ 𝑧 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) |
63 |
62
|
ralrimiva |
⊢ ( Lim 𝑧 → ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) |
64 |
|
df-sbc |
⊢ ( [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ 𝑧 ∈ { 𝑥 ∣ ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } ) |
65 |
52
|
eleq2i |
⊢ ( 𝑧 ∈ { 𝑥 ∣ ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) } ↔ 𝑧 ∈ { 𝑧 ∣ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } ) |
66 |
64 65
|
bitri |
⊢ ( [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ 𝑧 ∈ { 𝑧 ∣ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } ) |
67 |
|
abid |
⊢ ( 𝑧 ∈ { 𝑧 ∣ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) } ↔ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) |
68 |
66 67
|
bitri |
⊢ ( [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑧 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑧 ) ) |
69 |
63 68
|
sylibr |
⊢ ( Lim 𝑧 → [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) |
70 |
69
|
a1d |
⊢ ( Lim 𝑧 → ( ∀ 𝑥 ∈ 𝑧 ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → [ 𝑧 / 𝑥 ] ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) |
71 |
8 55 70
|
tfindes |
⊢ ( 𝑥 ∈ On → ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) |
72 |
|
rsp |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) → ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) |
73 |
71 72
|
syl |
⊢ ( 𝑥 ∈ On → ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) |
74 |
|
eleq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∈ On ↔ 𝑋 ∈ On ) ) |
75 |
74
|
adantl |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( 𝑥 ∈ On ↔ 𝑋 ∈ On ) ) |
76 |
|
eleq12 |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( 𝑦 ∈ 𝑥 ↔ 𝑌 ∈ 𝑋 ) ) |
77 |
|
fveq2 |
⊢ ( 𝑦 = 𝑌 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ) |
78 |
77
|
adantr |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ) |
79 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) |
80 |
79
|
adantl |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) = ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) |
81 |
78 80
|
sseq12d |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ↔ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) |
82 |
76 81
|
imbi12d |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ↔ ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) |
83 |
75 82
|
imbi12d |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( ( 𝑥 ∈ On → ( 𝑦 ∈ 𝑥 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑦 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑥 ) ) ) ↔ ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) ) |
84 |
73 83
|
mpbii |
⊢ ( ( 𝑦 = 𝑌 ∧ 𝑥 = 𝑋 ) → ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) |
85 |
84
|
ex |
⊢ ( 𝑦 = 𝑌 → ( 𝑥 = 𝑋 → ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) ) |
86 |
85
|
vtocleg |
⊢ ( 𝑌 ∈ 𝑋 → ( 𝑥 = 𝑋 → ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) ) |
87 |
86
|
com12 |
⊢ ( 𝑥 = 𝑋 → ( 𝑌 ∈ 𝑋 → ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) ) |
88 |
87
|
vtocleg |
⊢ ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) ) |
89 |
88
|
pm2.43b |
⊢ ( 𝑌 ∈ 𝑋 → ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) ) |
90 |
89
|
pm2.43b |
⊢ ( 𝑋 ∈ On → ( 𝑌 ∈ 𝑋 → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) ) |
91 |
90
|
imp |
⊢ ( ( 𝑋 ∈ On ∧ 𝑌 ∈ 𝑋 ) → ( rec ( 𝐹 , 𝐴 ) ‘ 𝑌 ) ⊆ ( rec ( 𝐹 , 𝐴 ) ‘ 𝑋 ) ) |