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 |
|
ordsson |
⊢ ( Ord dom 𝑅1 → dom 𝑅1 ⊆ On ) |
5 |
2 3 4
|
mp2b |
⊢ dom 𝑅1 ⊆ On |
6 |
5
|
sseli |
⊢ ( 𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = ∅ → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ ∅ ) ) |
8 |
|
r10 |
⊢ ( 𝑅1 ‘ ∅ ) = ∅ |
9 |
7 8
|
eqtrdi |
⊢ ( 𝑥 = ∅ → ( 𝑅1 ‘ 𝑥 ) = ∅ ) |
10 |
|
treq |
⊢ ( ( 𝑅1 ‘ 𝑥 ) = ∅ → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∅ ) ) |
11 |
9 10
|
syl |
⊢ ( 𝑥 = ∅ → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∅ ) ) |
12 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ 𝑦 ) ) |
13 |
|
treq |
⊢ ( ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ 𝑦 ) → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ( 𝑅1 ‘ 𝑦 ) ) ) |
14 |
12 13
|
syl |
⊢ ( 𝑥 = 𝑦 → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ( 𝑅1 ‘ 𝑦 ) ) ) |
15 |
|
fveq2 |
⊢ ( 𝑥 = suc 𝑦 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ suc 𝑦 ) ) |
16 |
|
treq |
⊢ ( ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ suc 𝑦 ) → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ( 𝑅1 ‘ suc 𝑦 ) ) ) |
17 |
15 16
|
syl |
⊢ ( 𝑥 = suc 𝑦 → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ( 𝑅1 ‘ suc 𝑦 ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ 𝐴 ) ) |
19 |
|
treq |
⊢ ( ( 𝑅1 ‘ 𝑥 ) = ( 𝑅1 ‘ 𝐴 ) → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ( 𝑅1 ‘ 𝐴 ) ) ) |
20 |
18 19
|
syl |
⊢ ( 𝑥 = 𝐴 → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ( 𝑅1 ‘ 𝐴 ) ) ) |
21 |
|
tr0 |
⊢ Tr ∅ |
22 |
|
limsuc |
⊢ ( Lim dom 𝑅1 → ( 𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) ) |
23 |
2 22
|
ax-mp |
⊢ ( 𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1 ) |
24 |
|
simpr |
⊢ ( ( 𝑦 ∈ On ∧ Tr ( 𝑅1 ‘ 𝑦 ) ) → Tr ( 𝑅1 ‘ 𝑦 ) ) |
25 |
|
pwtr |
⊢ ( Tr ( 𝑅1 ‘ 𝑦 ) ↔ Tr 𝒫 ( 𝑅1 ‘ 𝑦 ) ) |
26 |
24 25
|
sylib |
⊢ ( ( 𝑦 ∈ On ∧ Tr ( 𝑅1 ‘ 𝑦 ) ) → Tr 𝒫 ( 𝑅1 ‘ 𝑦 ) ) |
27 |
|
r1sucg |
⊢ ( 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ suc 𝑦 ) = 𝒫 ( 𝑅1 ‘ 𝑦 ) ) |
28 |
|
treq |
⊢ ( ( 𝑅1 ‘ suc 𝑦 ) = 𝒫 ( 𝑅1 ‘ 𝑦 ) → ( Tr ( 𝑅1 ‘ suc 𝑦 ) ↔ Tr 𝒫 ( 𝑅1 ‘ 𝑦 ) ) ) |
29 |
27 28
|
syl |
⊢ ( 𝑦 ∈ dom 𝑅1 → ( Tr ( 𝑅1 ‘ suc 𝑦 ) ↔ Tr 𝒫 ( 𝑅1 ‘ 𝑦 ) ) ) |
30 |
26 29
|
syl5ibrcom |
⊢ ( ( 𝑦 ∈ On ∧ Tr ( 𝑅1 ‘ 𝑦 ) ) → ( 𝑦 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ suc 𝑦 ) ) ) |
31 |
23 30
|
syl5bir |
⊢ ( ( 𝑦 ∈ On ∧ Tr ( 𝑅1 ‘ 𝑦 ) ) → ( suc 𝑦 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ suc 𝑦 ) ) ) |
32 |
|
ndmfv |
⊢ ( ¬ suc 𝑦 ∈ dom 𝑅1 → ( 𝑅1 ‘ suc 𝑦 ) = ∅ ) |
33 |
|
treq |
⊢ ( ( 𝑅1 ‘ suc 𝑦 ) = ∅ → ( Tr ( 𝑅1 ‘ suc 𝑦 ) ↔ Tr ∅ ) ) |
34 |
32 33
|
syl |
⊢ ( ¬ suc 𝑦 ∈ dom 𝑅1 → ( Tr ( 𝑅1 ‘ suc 𝑦 ) ↔ Tr ∅ ) ) |
35 |
21 34
|
mpbiri |
⊢ ( ¬ suc 𝑦 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ suc 𝑦 ) ) |
36 |
31 35
|
pm2.61d1 |
⊢ ( ( 𝑦 ∈ On ∧ Tr ( 𝑅1 ‘ 𝑦 ) ) → Tr ( 𝑅1 ‘ suc 𝑦 ) ) |
37 |
36
|
ex |
⊢ ( 𝑦 ∈ On → ( Tr ( 𝑅1 ‘ 𝑦 ) → Tr ( 𝑅1 ‘ suc 𝑦 ) ) ) |
38 |
|
triun |
⊢ ( ∀ 𝑦 ∈ 𝑥 Tr ( 𝑅1 ‘ 𝑦 ) → Tr ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) |
39 |
|
r1limg |
⊢ ( ( 𝑥 ∈ dom 𝑅1 ∧ Lim 𝑥 ) → ( 𝑅1 ‘ 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) |
40 |
39
|
ancoms |
⊢ ( ( Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → ( 𝑅1 ‘ 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) |
41 |
|
treq |
⊢ ( ( 𝑅1 ‘ 𝑥 ) = ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) ) |
42 |
40 41
|
syl |
⊢ ( ( Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∪ 𝑦 ∈ 𝑥 ( 𝑅1 ‘ 𝑦 ) ) ) |
43 |
38 42
|
syl5ibr |
⊢ ( ( Lim 𝑥 ∧ 𝑥 ∈ dom 𝑅1 ) → ( ∀ 𝑦 ∈ 𝑥 Tr ( 𝑅1 ‘ 𝑦 ) → Tr ( 𝑅1 ‘ 𝑥 ) ) ) |
44 |
43
|
impancom |
⊢ ( ( Lim 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 Tr ( 𝑅1 ‘ 𝑦 ) ) → ( 𝑥 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ 𝑥 ) ) ) |
45 |
|
ndmfv |
⊢ ( ¬ 𝑥 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝑥 ) = ∅ ) |
46 |
45 10
|
syl |
⊢ ( ¬ 𝑥 ∈ dom 𝑅1 → ( Tr ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∅ ) ) |
47 |
21 46
|
mpbiri |
⊢ ( ¬ 𝑥 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ 𝑥 ) ) |
48 |
44 47
|
pm2.61d1 |
⊢ ( ( Lim 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 Tr ( 𝑅1 ‘ 𝑦 ) ) → Tr ( 𝑅1 ‘ 𝑥 ) ) |
49 |
48
|
ex |
⊢ ( Lim 𝑥 → ( ∀ 𝑦 ∈ 𝑥 Tr ( 𝑅1 ‘ 𝑦 ) → Tr ( 𝑅1 ‘ 𝑥 ) ) ) |
50 |
11 14 17 20 21 37 49
|
tfinds |
⊢ ( 𝐴 ∈ On → Tr ( 𝑅1 ‘ 𝐴 ) ) |
51 |
6 50
|
syl |
⊢ ( 𝐴 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ 𝐴 ) ) |
52 |
|
ndmfv |
⊢ ( ¬ 𝐴 ∈ dom 𝑅1 → ( 𝑅1 ‘ 𝐴 ) = ∅ ) |
53 |
|
treq |
⊢ ( ( 𝑅1 ‘ 𝐴 ) = ∅ → ( Tr ( 𝑅1 ‘ 𝐴 ) ↔ Tr ∅ ) ) |
54 |
52 53
|
syl |
⊢ ( ¬ 𝐴 ∈ dom 𝑅1 → ( Tr ( 𝑅1 ‘ 𝐴 ) ↔ Tr ∅ ) ) |
55 |
21 54
|
mpbiri |
⊢ ( ¬ 𝐴 ∈ dom 𝑅1 → Tr ( 𝑅1 ‘ 𝐴 ) ) |
56 |
51 55
|
pm2.61i |
⊢ Tr ( 𝑅1 ‘ 𝐴 ) |