Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) |
2 |
|
relexpnndm |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ) → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ dom 𝑅 ) |
3 |
|
ssun1 |
⊢ dom 𝑅 ⊆ ( dom 𝑅 ∪ ran 𝑅 ) |
4 |
2 3
|
sstrdi |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉 ) → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) |
5 |
4
|
ex |
⊢ ( 𝑁 ∈ ℕ → ( 𝑅 ∈ 𝑉 → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
6 |
|
simpl |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → 𝑁 = 0 ) |
7 |
6
|
oveq2d |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → ( 𝑅 ↑𝑟 𝑁 ) = ( 𝑅 ↑𝑟 0 ) ) |
8 |
|
relexp0g |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝑅 ↑𝑟 0 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → ( 𝑅 ↑𝑟 0 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
10 |
7 9
|
eqtrd |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → ( 𝑅 ↑𝑟 𝑁 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
11 |
10
|
dmeqd |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → dom ( 𝑅 ↑𝑟 𝑁 ) = dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
12 |
|
dmresi |
⊢ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
13 |
11 12
|
eqtrdi |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → dom ( 𝑅 ↑𝑟 𝑁 ) = ( dom 𝑅 ∪ ran 𝑅 ) ) |
14 |
|
eqimss |
⊢ ( dom ( 𝑅 ↑𝑟 𝑁 ) = ( dom 𝑅 ∪ ran 𝑅 ) → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) |
15 |
13 14
|
syl |
⊢ ( ( 𝑁 = 0 ∧ 𝑅 ∈ 𝑉 ) → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) |
16 |
15
|
ex |
⊢ ( 𝑁 = 0 → ( 𝑅 ∈ 𝑉 → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
17 |
5 16
|
jaoi |
⊢ ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) → ( 𝑅 ∈ 𝑉 → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
18 |
1 17
|
sylbi |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑅 ∈ 𝑉 → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
19 |
18
|
imp |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ) → dom ( 𝑅 ↑𝑟 𝑁 ) ⊆ ( dom 𝑅 ∪ ran 𝑅 ) ) |