Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
2 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
3 |
|
relexpcnv |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝑅 ∈ V ) → ◡ ( 𝑅 ↑𝑟 𝑛 ) = ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
4 |
2 3
|
sylan |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑅 ∈ V ) → ◡ ( 𝑅 ↑𝑟 𝑛 ) = ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
5 |
4
|
expcom |
⊢ ( 𝑅 ∈ V → ( 𝑛 ∈ ℕ → ◡ ( 𝑅 ↑𝑟 𝑛 ) = ( ◡ 𝑅 ↑𝑟 𝑛 ) ) ) |
6 |
5
|
ralrimiv |
⊢ ( 𝑅 ∈ V → ∀ 𝑛 ∈ ℕ ◡ ( 𝑅 ↑𝑟 𝑛 ) = ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
7 |
|
iuneq2 |
⊢ ( ∀ 𝑛 ∈ ℕ ◡ ( 𝑅 ↑𝑟 𝑛 ) = ( ◡ 𝑅 ↑𝑟 𝑛 ) → ∪ 𝑛 ∈ ℕ ◡ ( 𝑅 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
8 |
6 7
|
syl |
⊢ ( 𝑅 ∈ V → ∪ 𝑛 ∈ ℕ ◡ ( 𝑅 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
9 |
|
oveq1 |
⊢ ( 𝑟 = 𝑅 → ( 𝑟 ↑𝑟 𝑛 ) = ( 𝑅 ↑𝑟 𝑛 ) ) |
10 |
9
|
iuneq2d |
⊢ ( 𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ ( 𝑟 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ ( 𝑅 ↑𝑟 𝑛 ) ) |
11 |
|
dftrcl3 |
⊢ t+ = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ ( 𝑟 ↑𝑟 𝑛 ) ) |
12 |
|
nnex |
⊢ ℕ ∈ V |
13 |
|
ovex |
⊢ ( 𝑅 ↑𝑟 𝑛 ) ∈ V |
14 |
12 13
|
iunex |
⊢ ∪ 𝑛 ∈ ℕ ( 𝑅 ↑𝑟 𝑛 ) ∈ V |
15 |
10 11 14
|
fvmpt |
⊢ ( 𝑅 ∈ V → ( t+ ‘ 𝑅 ) = ∪ 𝑛 ∈ ℕ ( 𝑅 ↑𝑟 𝑛 ) ) |
16 |
15
|
cnveqd |
⊢ ( 𝑅 ∈ V → ◡ ( t+ ‘ 𝑅 ) = ◡ ∪ 𝑛 ∈ ℕ ( 𝑅 ↑𝑟 𝑛 ) ) |
17 |
|
cnviun |
⊢ ◡ ∪ 𝑛 ∈ ℕ ( 𝑅 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ ◡ ( 𝑅 ↑𝑟 𝑛 ) |
18 |
16 17
|
eqtrdi |
⊢ ( 𝑅 ∈ V → ◡ ( t+ ‘ 𝑅 ) = ∪ 𝑛 ∈ ℕ ◡ ( 𝑅 ↑𝑟 𝑛 ) ) |
19 |
|
cnvexg |
⊢ ( 𝑅 ∈ V → ◡ 𝑅 ∈ V ) |
20 |
|
oveq1 |
⊢ ( 𝑠 = ◡ 𝑅 → ( 𝑠 ↑𝑟 𝑛 ) = ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
21 |
20
|
iuneq2d |
⊢ ( 𝑠 = ◡ 𝑅 → ∪ 𝑛 ∈ ℕ ( 𝑠 ↑𝑟 𝑛 ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
22 |
|
dftrcl3 |
⊢ t+ = ( 𝑠 ∈ V ↦ ∪ 𝑛 ∈ ℕ ( 𝑠 ↑𝑟 𝑛 ) ) |
23 |
|
ovex |
⊢ ( ◡ 𝑅 ↑𝑟 𝑛 ) ∈ V |
24 |
12 23
|
iunex |
⊢ ∪ 𝑛 ∈ ℕ ( ◡ 𝑅 ↑𝑟 𝑛 ) ∈ V |
25 |
21 22 24
|
fvmpt |
⊢ ( ◡ 𝑅 ∈ V → ( t+ ‘ ◡ 𝑅 ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
26 |
19 25
|
syl |
⊢ ( 𝑅 ∈ V → ( t+ ‘ ◡ 𝑅 ) = ∪ 𝑛 ∈ ℕ ( ◡ 𝑅 ↑𝑟 𝑛 ) ) |
27 |
8 18 26
|
3eqtr4d |
⊢ ( 𝑅 ∈ V → ◡ ( t+ ‘ 𝑅 ) = ( t+ ‘ ◡ 𝑅 ) ) |
28 |
1 27
|
syl |
⊢ ( 𝑅 ∈ 𝑉 → ◡ ( t+ ‘ 𝑅 ) = ( t+ ‘ ◡ 𝑅 ) ) |