Step |
Hyp |
Ref |
Expression |
1 |
|
r1funlim |
⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 ) |
2 |
1
|
simpri |
⊢ Lim dom 𝑅1 |
3 |
|
limord |
⊢ ( Lim dom 𝑅1 → Ord dom 𝑅1 ) |
4 |
2 3
|
ax-mp |
⊢ Ord dom 𝑅1 |
5 |
|
ordelon |
⊢ ( ( Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1 ) → 𝐴 ∈ On ) |
6 |
4 5
|
mpan |
⊢ ( 𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On ) |
7 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1 ) ) |
8 |
|
eleq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ↔ 𝑦 ∈ ∪ ( 𝑅1 “ On ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( rank ‘ 𝑥 ) = ( rank ‘ 𝑦 ) ) |
10 |
|
id |
⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 ) |
11 |
9 10
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( rank ‘ 𝑥 ) = 𝑥 ↔ ( rank ‘ 𝑦 ) = 𝑦 ) ) |
12 |
8 11
|
anbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ) |
13 |
7 12
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ dom 𝑅1 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) ↔ ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ) ) |
14 |
|
eleq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1 ) ) |
15 |
|
eleq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ↔ 𝐴 ∈ ∪ ( 𝑅1 “ On ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( rank ‘ 𝑥 ) = ( rank ‘ 𝐴 ) ) |
17 |
|
id |
⊢ ( 𝑥 = 𝐴 → 𝑥 = 𝐴 ) |
18 |
16 17
|
eqeq12d |
⊢ ( 𝑥 = 𝐴 → ( ( rank ‘ 𝑥 ) = 𝑥 ↔ ( rank ‘ 𝐴 ) = 𝐴 ) ) |
19 |
15 18
|
anbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝐴 ) = 𝐴 ) ) ) |
20 |
14 19
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ dom 𝑅1 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) ↔ ( 𝐴 ∈ dom 𝑅1 → ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝐴 ) = 𝐴 ) ) ) ) |
21 |
|
ordtr1 |
⊢ ( Ord dom 𝑅1 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → 𝑦 ∈ dom 𝑅1 ) ) |
22 |
4 21
|
ax-mp |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → 𝑦 ∈ dom 𝑅1 ) |
23 |
22
|
ancoms |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ dom 𝑅1 ) |
24 |
|
pm5.5 |
⊢ ( 𝑦 ∈ dom 𝑅1 → ( ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ↔ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ) |
25 |
23 24
|
syl |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ↔ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ) |
26 |
25
|
ralbidva |
⊢ ( 𝑥 ∈ dom 𝑅1 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) ) |
27 |
|
simplr |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑦 ∈ 𝑥 ) |
28 |
|
ordelon |
⊢ ( ( Ord dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1 ) → 𝑥 ∈ On ) |
29 |
4 28
|
mpan |
⊢ ( 𝑥 ∈ dom 𝑅1 → 𝑥 ∈ On ) |
30 |
29
|
ad2antrr |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ On ) |
31 |
|
eloni |
⊢ ( 𝑥 ∈ On → Ord 𝑥 ) |
32 |
30 31
|
syl |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → Ord 𝑥 ) |
33 |
|
ordelsuc |
⊢ ( ( 𝑦 ∈ 𝑥 ∧ Ord 𝑥 ) → ( 𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥 ) ) |
34 |
27 32 33
|
syl2anc |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥 ) ) |
35 |
27 34
|
mpbid |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → suc 𝑦 ⊆ 𝑥 ) |
36 |
23
|
adantr |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑦 ∈ dom 𝑅1 ) |
37 |
|
limsuc |
⊢ ( Lim dom 𝑅1 → ( 𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) ) |
38 |
2 37
|
ax-mp |
⊢ ( 𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) |
39 |
36 38
|
sylib |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → suc 𝑦 ∈ dom 𝑅1 ) |
40 |
|
simpll |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ dom 𝑅1 ) |
41 |
|
r1ord3g |
⊢ ( ( suc 𝑦 ∈ dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1 ) → ( suc 𝑦 ⊆ 𝑥 → ( 𝑅1 ‘ suc 𝑦 ) ⊆ ( 𝑅1 ‘ 𝑥 ) ) ) |
42 |
39 40 41
|
syl2anc |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( suc 𝑦 ⊆ 𝑥 → ( 𝑅1 ‘ suc 𝑦 ) ⊆ ( 𝑅1 ‘ 𝑥 ) ) ) |
43 |
35 42
|
mpd |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑅1 ‘ suc 𝑦 ) ⊆ ( 𝑅1 ‘ 𝑥 ) ) |
44 |
|
rankidb |
⊢ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) → 𝑦 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝑦 ) ) ) |
45 |
44
|
ad2antrl |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑦 ∈ ( 𝑅1 ‘ suc ( rank ‘ 𝑦 ) ) ) |
46 |
|
suceq |
⊢ ( ( rank ‘ 𝑦 ) = 𝑦 → suc ( rank ‘ 𝑦 ) = suc 𝑦 ) |
47 |
46
|
ad2antll |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → suc ( rank ‘ 𝑦 ) = suc 𝑦 ) |
48 |
47
|
fveq2d |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑅1 ‘ suc ( rank ‘ 𝑦 ) ) = ( 𝑅1 ‘ suc 𝑦 ) ) |
49 |
45 48
|
eleqtrd |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑦 ∈ ( 𝑅1 ‘ suc 𝑦 ) ) |
50 |
43 49
|
sseldd |
⊢ ( ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) ∧ ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) ) |
51 |
50
|
ex |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) ) ) |
52 |
51
|
ralimdva |
⊢ ( 𝑥 ∈ dom 𝑅1 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) ) ) |
53 |
52
|
imp |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) ) |
54 |
|
dfss3 |
⊢ ( 𝑥 ⊆ ( 𝑅1 ‘ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ ( 𝑅1 ‘ 𝑥 ) ) |
55 |
53 54
|
sylibr |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ⊆ ( 𝑅1 ‘ 𝑥 ) ) |
56 |
|
vex |
⊢ 𝑥 ∈ V |
57 |
56
|
elpw |
⊢ ( 𝑥 ∈ 𝒫 ( 𝑅1 ‘ 𝑥 ) ↔ 𝑥 ⊆ ( 𝑅1 ‘ 𝑥 ) ) |
58 |
55 57
|
sylibr |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ 𝒫 ( 𝑅1 ‘ 𝑥 ) ) |
59 |
|
r1sucg |
⊢ ( 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ suc 𝑥 ) = 𝒫 ( 𝑅1 ‘ 𝑥 ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑅1 ‘ suc 𝑥 ) = 𝒫 ( 𝑅1 ‘ 𝑥 ) ) |
61 |
58 60
|
eleqtrrd |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ ( 𝑅1 ‘ suc 𝑥 ) ) |
62 |
|
r1elwf |
⊢ ( 𝑥 ∈ ( 𝑅1 ‘ suc 𝑥 ) → 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
63 |
61 62
|
syl |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
64 |
|
rankval3b |
⊢ ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝑥 ) = ∩ { 𝑧 ∈ On ∣ ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 } ) |
65 |
63 64
|
syl |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( rank ‘ 𝑥 ) = ∩ { 𝑧 ∈ On ∣ ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 } ) |
66 |
|
eleq1 |
⊢ ( ( rank ‘ 𝑦 ) = 𝑦 → ( ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) ) |
67 |
66
|
adantl |
⊢ ( ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ( ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) ) |
68 |
67
|
ralimi |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ∀ 𝑦 ∈ 𝑥 ( ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) ) |
69 |
|
ralbi |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) → ( ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝑧 ) ) |
70 |
68 69
|
syl |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ( ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝑧 ) ) |
71 |
|
dfss3 |
⊢ ( 𝑥 ⊆ 𝑧 ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝑧 ) |
72 |
70 71
|
bitr4di |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ( ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧 ) ) |
73 |
72
|
rabbidv |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → { 𝑧 ∈ On ∣ ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 } = { 𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧 } ) |
74 |
73
|
inteqd |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ∩ { 𝑧 ∈ On ∣ ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 } = ∩ { 𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧 } ) |
75 |
74
|
adantl |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ∩ { 𝑧 ∈ On ∣ ∀ 𝑦 ∈ 𝑥 ( rank ‘ 𝑦 ) ∈ 𝑧 } = ∩ { 𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧 } ) |
76 |
29
|
adantr |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → 𝑥 ∈ On ) |
77 |
|
intmin |
⊢ ( 𝑥 ∈ On → ∩ { 𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧 } = 𝑥 ) |
78 |
76 77
|
syl |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ∩ { 𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧 } = 𝑥 ) |
79 |
65 75 78
|
3eqtrd |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( rank ‘ 𝑥 ) = 𝑥 ) |
80 |
63 79
|
jca |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) |
81 |
80
|
ex |
⊢ ( 𝑥 ∈ dom 𝑅1 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) ) |
82 |
26 81
|
sylbid |
⊢ ( 𝑥 ∈ dom 𝑅1 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) ) |
83 |
82
|
com12 |
⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑥 ∈ dom 𝑅1 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) ) |
84 |
83
|
a1i |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( 𝑦 ∈ dom 𝑅1 → ( 𝑦 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑦 ) = 𝑦 ) ) → ( 𝑥 ∈ dom 𝑅1 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) ) ) |
85 |
13 20 84
|
tfis3 |
⊢ ( 𝐴 ∈ On → ( 𝐴 ∈ dom 𝑅1 → ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝐴 ) = 𝐴 ) ) ) |
86 |
6 85
|
mpcom |
⊢ ( 𝐴 ∈ dom 𝑅1 → ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝐴 ) = 𝐴 ) ) |