Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵 ) → 𝐵 ∈ dom 𝑅1 ) |
2 |
|
r1funlim |
⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 ) |
3 |
2
|
simpri |
⊢ Lim dom 𝑅1 |
4 |
|
limord |
⊢ ( Lim dom 𝑅1 → Ord dom 𝑅1 ) |
5 |
3 4
|
ax-mp |
⊢ Ord dom 𝑅1 |
6 |
|
ordsson |
⊢ ( Ord dom 𝑅1 → dom 𝑅1 ⊆ On ) |
7 |
5 6
|
ax-mp |
⊢ dom 𝑅1 ⊆ On |
8 |
7
|
sseli |
⊢ ( 𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On ) |
9 |
1 8
|
syl |
⊢ ( ( 𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵 ) → 𝐵 ∈ On ) |
10 |
|
onelon |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ 𝐵 ) → 𝐴 ∈ On ) |
11 |
8 10
|
sylan |
⊢ ( ( 𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵 ) → 𝐴 ∈ On ) |
12 |
|
suceloni |
⊢ ( 𝐴 ∈ On → suc 𝐴 ∈ On ) |
13 |
11 12
|
syl |
⊢ ( ( 𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵 ) → suc 𝐴 ∈ On ) |
14 |
|
eloni |
⊢ ( 𝐵 ∈ On → Ord 𝐵 ) |
15 |
|
ordsucss |
⊢ ( Ord 𝐵 → ( 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵 ) ) |
16 |
14 15
|
syl |
⊢ ( 𝐵 ∈ On → ( 𝐴 ∈ 𝐵 → suc 𝐴 ⊆ 𝐵 ) ) |
17 |
16
|
imp |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ 𝐵 ) → suc 𝐴 ⊆ 𝐵 ) |
18 |
8 17
|
sylan |
⊢ ( ( 𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵 ) → suc 𝐴 ⊆ 𝐵 ) |
19 |
|
eleq1 |
⊢ ( 𝑥 = suc 𝐴 → ( 𝑥 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1 ) ) |
20 |
|
fveq2 |
⊢ ( 𝑥 = suc 𝐴 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ suc 𝐴 ) ) |
21 |
20
|
eleq2d |
⊢ ( 𝑥 = suc 𝐴 → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ↔ ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) ) |
22 |
19 21
|
imbi12d |
⊢ ( 𝑥 = suc 𝐴 → ( ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) ↔ ( suc 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) ) ) |
23 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1 ) ) |
24 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ 𝑦 ) ) |
25 |
24
|
eleq2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ↔ ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) ) |
26 |
23 25
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) ↔ ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) ) ) |
27 |
|
eleq1 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝑥 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) ) |
28 |
|
fveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ suc 𝑦 ) ) |
29 |
28
|
eleq2d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ↔ ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝑦 ) ) ) |
30 |
27 29
|
imbi12d |
⊢ ( 𝑥 = suc 𝑦 → ( ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) ↔ ( suc 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝑦 ) ) ) ) |
31 |
|
eleq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ∈ dom 𝑅1 ↔ 𝐵 ∈ dom 𝑅1 ) ) |
32 |
|
fveq2 |
⊢ ( 𝑥 = 𝐵 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ 𝐵 ) ) |
33 |
32
|
eleq2d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ↔ ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) ) |
34 |
31 33
|
imbi12d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) ↔ ( 𝐵 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) ) ) |
35 |
|
fvex |
⊢ ( 𝑅1 ‘ 𝐴 ) ∈ V |
36 |
35
|
pwid |
⊢ ( 𝑅1 ‘ 𝐴 ) ∈ 𝒫 ( 𝑅1 ‘ 𝐴 ) |
37 |
|
limsuc |
⊢ ( Lim dom 𝑅1 → ( 𝐴 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1 ) ) |
38 |
3 37
|
ax-mp |
⊢ ( 𝐴 ∈ dom 𝑅1 ↔ suc 𝐴 ∈ dom 𝑅1 ) |
39 |
|
r1sucg |
⊢ ( 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ suc 𝐴 ) = 𝒫 ( 𝑅1 ‘ 𝐴 ) ) |
40 |
38 39
|
sylbir |
⊢ ( suc 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ suc 𝐴 ) = 𝒫 ( 𝑅1 ‘ 𝐴 ) ) |
41 |
36 40
|
eleqtrrid |
⊢ ( suc 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) |
42 |
41
|
a1i |
⊢ ( suc 𝐴 ∈ On → ( suc 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) ) |
43 |
|
limsuc |
⊢ ( Lim dom 𝑅1 → ( 𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) ) |
44 |
3 43
|
ax-mp |
⊢ ( 𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) |
45 |
|
r1tr |
⊢ Tr ( 𝑅1 ‘ 𝑦 ) |
46 |
|
dftr4 |
⊢ ( Tr ( 𝑅1 ‘ 𝑦 ) ↔ ( 𝑅1 ‘ 𝑦 ) ⊆ 𝒫 ( 𝑅1 ‘ 𝑦 ) ) |
47 |
45 46
|
mpbi |
⊢ ( 𝑅1 ‘ 𝑦 ) ⊆ 𝒫 ( 𝑅1 ‘ 𝑦 ) |
48 |
|
r1sucg |
⊢ ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ suc 𝑦 ) = 𝒫 ( 𝑅1 ‘ 𝑦 ) ) |
49 |
47 48
|
sseqtrrid |
⊢ ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝑦 ) ⊆ ( 𝑅1 ‘ suc 𝑦 ) ) |
50 |
49
|
sseld |
⊢ ( 𝑦 ∈ dom 𝑅1 → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝑦 ) ) ) |
51 |
50
|
a2i |
⊢ ( ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) → ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝑦 ) ) ) |
52 |
44 51
|
syl5bir |
⊢ ( ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) → ( suc 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝑦 ) ) ) |
53 |
52
|
a1i |
⊢ ( ( ( 𝑦 ∈ On ∧ suc 𝐴 ∈ On ) ∧ suc 𝐴 ⊆ 𝑦 ) → ( ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) → ( suc 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝑦 ) ) ) ) |
54 |
|
simprl |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → suc 𝐴 ⊆ 𝑥 ) |
55 |
|
simplr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → suc 𝐴 ∈ On ) |
56 |
|
sucelon |
⊢ ( 𝐴 ∈ On ↔ suc 𝐴 ∈ On ) |
57 |
55 56
|
sylibr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → 𝐴 ∈ On ) |
58 |
|
limord |
⊢ ( Lim 𝑥 → Ord 𝑥 ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → Ord 𝑥 ) |
60 |
|
ordelsuc |
⊢ ( ( 𝐴 ∈ On ∧ Ord 𝑥 ) → ( 𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥 ) ) |
61 |
57 59 60
|
syl2anc |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥 ) ) |
62 |
54 61
|
mpbird |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → 𝐴 ∈ 𝑥 ) |
63 |
|
limsuc |
⊢ ( Lim 𝑥 → ( 𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥 ) ) |
64 |
63
|
ad2antrr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝐴 ∈ 𝑥 ↔ suc 𝐴 ∈ 𝑥 ) ) |
65 |
62 64
|
mpbid |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → suc 𝐴 ∈ 𝑥 ) |
66 |
|
simprr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → 𝑥 ∈ dom 𝑅1 ) |
67 |
|
ordtr1 |
⊢ ( Ord dom 𝑅1 → ( ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → 𝐴 ∈ dom 𝑅1 ) ) |
68 |
5 67
|
ax-mp |
⊢ ( ( 𝐴 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → 𝐴 ∈ dom 𝑅1 ) |
69 |
62 66 68
|
syl2anc |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → 𝐴 ∈ dom 𝑅1 ) |
70 |
69 39
|
syl |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝑅1 ‘ suc 𝐴 ) = 𝒫 ( 𝑅1 ‘ 𝐴 ) ) |
71 |
36 70
|
eleqtrrid |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) |
72 |
|
fveq2 |
⊢ ( 𝑦 = suc 𝐴 → ( 𝑅1 ‘ 𝑦 ) = ( 𝑅1 ‘ suc 𝐴 ) ) |
73 |
72
|
eleq2d |
⊢ ( 𝑦 = suc 𝐴 → ( ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ↔ ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) ) |
74 |
73
|
rspcev |
⊢ ( ( suc 𝐴 ∈ 𝑥 ∧ ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ suc 𝐴 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) |
75 |
65 71 74
|
syl2anc |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ∃ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) |
76 |
|
eliun |
⊢ ( ( 𝑅1 ‘ 𝐴 ) ∈ ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ↔ ∃ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) |
77 |
75 76
|
sylibr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝑅1 ‘ 𝐴 ) ∈ ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) |
78 |
|
simpll |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → Lim 𝑥 ) |
79 |
|
r1limg |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥 ) → ( 𝑅1 ‘ 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) |
80 |
66 78 79
|
syl2anc |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝑅1 ‘ 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) |
81 |
77 80
|
eleqtrrd |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) ) → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) |
82 |
81
|
expr |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ suc 𝐴 ⊆ 𝑥 ) → ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) ) |
83 |
82
|
a1d |
⊢ ( ( ( Lim 𝑥 ∧ suc 𝐴 ∈ On ) ∧ suc 𝐴 ⊆ 𝑥 ) → ( ∀ 𝑦 ∈ 𝑥 ( suc 𝐴 ⊆ 𝑦 → ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑦 ) ) ) → ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝑥 ) ) ) ) |
84 |
22 26 30 34 42 53 83
|
tfindsg |
⊢ ( ( ( 𝐵 ∈ On ∧ suc 𝐴 ∈ On ) ∧ suc 𝐴 ⊆ 𝐵 ) → ( 𝐵 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) ) |
85 |
84
|
impr |
⊢ ( ( ( 𝐵 ∈ On ∧ suc 𝐴 ∈ On ) ∧ ( suc 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ dom 𝑅1 ) ) → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) |
86 |
9 13 18 1 85
|
syl22anc |
⊢ ( ( 𝐵 ∈ dom 𝑅1 ∧ 𝐴 ∈ 𝐵 ) → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) |
87 |
86
|
ex |
⊢ ( 𝐵 ∈ dom 𝑅1 → ( 𝐴 ∈ 𝐵 → ( 𝑅1 ‘ 𝐴 ) ∈ ( 𝑅1 ‘ 𝐵 ) ) ) |