Step |
Hyp |
Ref |
Expression |
1 |
|
brfvidRP.r |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
2 |
|
dfid6 |
⊢ I = ( 𝑟 ∈ V ↦ ∪ 𝑛 ∈ { 1 } ( 𝑟 ↑𝑟 𝑛 ) ) |
3 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
4 |
|
snssi |
⊢ ( 1 ∈ ℕ0 → { 1 } ⊆ ℕ0 ) |
5 |
3 4
|
mp1i |
⊢ ( 𝜑 → { 1 } ⊆ ℕ0 ) |
6 |
2 1 5
|
brmptiunrelexpd |
⊢ ( 𝜑 → ( 𝐴 ( I ‘ 𝑅 ) 𝐵 ↔ ∃ 𝑛 ∈ { 1 } 𝐴 ( 𝑅 ↑𝑟 𝑛 ) 𝐵 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 𝑅 ↑𝑟 𝑛 ) = ( 𝑅 ↑𝑟 1 ) ) |
8 |
7
|
breqd |
⊢ ( 𝑛 = 1 → ( 𝐴 ( 𝑅 ↑𝑟 𝑛 ) 𝐵 ↔ 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ) ) |
9 |
8
|
rexsng |
⊢ ( 1 ∈ ℕ0 → ( ∃ 𝑛 ∈ { 1 } 𝐴 ( 𝑅 ↑𝑟 𝑛 ) 𝐵 ↔ 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ) ) |
10 |
3 9
|
mp1i |
⊢ ( 𝜑 → ( ∃ 𝑛 ∈ { 1 } 𝐴 ( 𝑅 ↑𝑟 𝑛 ) 𝐵 ↔ 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ) ) |
11 |
1
|
relexp1d |
⊢ ( 𝜑 → ( 𝑅 ↑𝑟 1 ) = 𝑅 ) |
12 |
11
|
breqd |
⊢ ( 𝜑 → ( 𝐴 ( 𝑅 ↑𝑟 1 ) 𝐵 ↔ 𝐴 𝑅 𝐵 ) ) |
13 |
6 10 12
|
3bitrd |
⊢ ( 𝜑 → ( 𝐴 ( I ‘ 𝑅 ) 𝐵 ↔ 𝐴 𝑅 𝐵 ) ) |