Step |
Hyp |
Ref |
Expression |
1 |
|
rtrclreclem1.1 |
⊢ ( 𝜑 → 𝑅 ∈ 𝑉 ) |
2 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
3 |
|
ssidd |
⊢ ( 𝜑 → 𝑅 ⊆ 𝑅 ) |
4 |
1
|
relexp1d |
⊢ ( 𝜑 → ( 𝑅 ↑𝑟 1 ) = 𝑅 ) |
5 |
3 4
|
sseqtrrd |
⊢ ( 𝜑 → 𝑅 ⊆ ( 𝑅 ↑𝑟 1 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 𝑅 ↑𝑟 𝑛 ) = ( 𝑅 ↑𝑟 1 ) ) |
7 |
6
|
sseq2d |
⊢ ( 𝑛 = 1 → ( 𝑅 ⊆ ( 𝑅 ↑𝑟 𝑛 ) ↔ 𝑅 ⊆ ( 𝑅 ↑𝑟 1 ) ) ) |
8 |
7
|
rspcev |
⊢ ( ( 1 ∈ ℕ0 ∧ 𝑅 ⊆ ( 𝑅 ↑𝑟 1 ) ) → ∃ 𝑛 ∈ ℕ0 𝑅 ⊆ ( 𝑅 ↑𝑟 𝑛 ) ) |
9 |
2 5 8
|
sylancr |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ℕ0 𝑅 ⊆ ( 𝑅 ↑𝑟 𝑛 ) ) |
10 |
|
ssiun |
⊢ ( ∃ 𝑛 ∈ ℕ0 𝑅 ⊆ ( 𝑅 ↑𝑟 𝑛 ) → 𝑅 ⊆ ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → 𝑅 ⊆ ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ) |
12 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ) |
13 |
|
oveq1 |
⊢ ( 𝑟 = 𝑅 → ( 𝑟 ↑𝑟 𝑛 ) = ( 𝑅 ↑𝑟 𝑛 ) ) |
14 |
13
|
iuneq2d |
⊢ ( 𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑟 = 𝑅 ) → ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ) |
16 |
1
|
elexd |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
17 |
|
nn0ex |
⊢ ℕ0 ∈ V |
18 |
|
ovex |
⊢ ( 𝑅 ↑𝑟 𝑛 ) ∈ V |
19 |
17 18
|
iunex |
⊢ ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ∈ V |
20 |
19
|
a1i |
⊢ ( 𝜑 → ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ∈ V ) |
21 |
12 15 16 20
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ‘ 𝑅 ) = ∪ 𝑛 ∈ ℕ0 ( 𝑅 ↑𝑟 𝑛 ) ) |
22 |
11 21
|
sseqtrrd |
⊢ ( 𝜑 → 𝑅 ⊆ ( ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ‘ 𝑅 ) ) |
23 |
|
df-rtrclrec |
⊢ t*rec = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) |
24 |
|
fveq1 |
⊢ ( t*rec = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) → ( t*rec ‘ 𝑅 ) = ( ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ‘ 𝑅 ) ) |
25 |
24
|
sseq2d |
⊢ ( t*rec = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) → ( 𝑅 ⊆ ( t*rec ‘ 𝑅 ) ↔ 𝑅 ⊆ ( ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ‘ 𝑅 ) ) ) |
26 |
25
|
imbi2d |
⊢ ( t*rec = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) → ( ( 𝜑 → 𝑅 ⊆ ( t*rec ‘ 𝑅 ) ) ↔ ( 𝜑 → 𝑅 ⊆ ( ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ‘ 𝑅 ) ) ) ) |
27 |
23 26
|
ax-mp |
⊢ ( ( 𝜑 → 𝑅 ⊆ ( t*rec ‘ 𝑅 ) ) ↔ ( 𝜑 → 𝑅 ⊆ ( ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 ( 𝑟 ↑𝑟 𝑛 ) ) ‘ 𝑅 ) ) ) |
28 |
22 27
|
mpbir |
⊢ ( 𝜑 → 𝑅 ⊆ ( t*rec ‘ 𝑅 ) ) |