Step |
Hyp |
Ref |
Expression |
1 |
|
brfvrcld2.r |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
2 |
1
|
brfvrcld |
⊢ ( 𝜑 → ( 𝐴 ( r* ‘ 𝑅 ) 𝐵 ↔ ( 𝐴 ( 𝑅 ↑𝑟 0 ) 𝐵 ∨ 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ) ) ) |
3 |
|
relexp0g |
⊢ ( 𝑅 ∈ V → ( 𝑅 ↑𝑟 0 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
4 |
1 3
|
syl |
⊢ ( 𝜑 → ( 𝑅 ↑𝑟 0 ) = ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
5 |
4
|
breqd |
⊢ ( 𝜑 → ( 𝐴 ( 𝑅 ↑𝑟 0 ) 𝐵 ↔ 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 ) ) |
6 |
|
relres |
⊢ Rel ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) |
7 |
6
|
releldmi |
⊢ ( 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 → 𝐴 ∈ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
8 |
6
|
relelrni |
⊢ ( 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 → 𝐵 ∈ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
9 |
|
dmresi |
⊢ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
10 |
9
|
eleq2i |
⊢ ( 𝐴 ∈ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ↔ 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) |
11 |
10
|
biimpi |
⊢ ( 𝐴 ∈ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) → 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) |
12 |
|
rnresi |
⊢ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) = ( dom 𝑅 ∪ ran 𝑅 ) |
13 |
12
|
eleq2i |
⊢ ( 𝐵 ∈ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ↔ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) |
14 |
13
|
biimpi |
⊢ ( 𝐵 ∈ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) → 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) |
15 |
11 14
|
anim12i |
⊢ ( ( 𝐴 ∈ dom ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ∧ 𝐵 ∈ ran ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) ) → ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
16 |
7 8 15
|
syl2anc |
⊢ ( 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 → ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) ) |
17 |
|
resieq |
⊢ ( ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) → ( 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 ↔ 𝐴 = 𝐵 ) ) |
18 |
16 17
|
biadanii |
⊢ ( 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 ↔ ( ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) ∧ 𝐴 = 𝐵 ) ) |
19 |
|
df-3an |
⊢ ( ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐴 = 𝐵 ) ↔ ( ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ) ∧ 𝐴 = 𝐵 ) ) |
20 |
18 19
|
bitr4i |
⊢ ( 𝐴 ( I ↾ ( dom 𝑅 ∪ ran 𝑅 ) ) 𝐵 ↔ ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐴 = 𝐵 ) ) |
21 |
5 20
|
bitrdi |
⊢ ( 𝜑 → ( 𝐴 ( 𝑅 ↑𝑟 0 ) 𝐵 ↔ ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐴 = 𝐵 ) ) ) |
22 |
1
|
relexp1d |
⊢ ( 𝜑 → ( 𝑅 ↑𝑟 1 ) = 𝑅 ) |
23 |
22
|
breqd |
⊢ ( 𝜑 → ( 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ↔ 𝐴 𝑅 𝐵 ) ) |
24 |
21 23
|
orbi12d |
⊢ ( 𝜑 → ( ( 𝐴 ( 𝑅 ↑𝑟 0 ) 𝐵 ∨ 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ) ↔ ( ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐴 = 𝐵 ) ∨ 𝐴 𝑅 𝐵 ) ) ) |
25 |
2 24
|
bitrd |
⊢ ( 𝜑 → ( 𝐴 ( r* ‘ 𝑅 ) 𝐵 ↔ ( ( 𝐴 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐵 ∈ ( dom 𝑅 ∪ ran 𝑅 ) ∧ 𝐴 = 𝐵 ) ∨ 𝐴 𝑅 𝐵 ) ) ) |