Step |
Hyp |
Ref |
Expression |
1 |
|
relcnv |
⊢ Rel ◡ 𝐹 |
2 |
|
id |
⊢ ( 𝑦 = 𝑧 → 𝑦 = 𝑧 ) |
3 |
2
|
inecmo |
⊢ ( Rel ◡ 𝐹 → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ) ) |
4 |
1 3
|
ax-mp |
⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ) |
5 |
|
brcnvg |
⊢ ( ( 𝑦 ∈ V ∧ 𝑥 ∈ V ) → ( 𝑦 ◡ 𝐹 𝑥 ↔ 𝑥 𝐹 𝑦 ) ) |
6 |
5
|
el2v |
⊢ ( 𝑦 ◡ 𝐹 𝑥 ↔ 𝑥 𝐹 𝑦 ) |
7 |
6
|
rmobii |
⊢ ( ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ↔ ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 ) |
8 |
7
|
albii |
⊢ ( ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑦 ◡ 𝐹 𝑥 ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 ) |
9 |
4 8
|
bitri |
⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( 𝑦 = 𝑧 ∨ ( [ 𝑦 ] ◡ 𝐹 ∩ [ 𝑧 ] ◡ 𝐹 ) = ∅ ) ↔ ∀ 𝑥 ∃* 𝑦 ∈ 𝐵 𝑥 𝐹 𝑦 ) |