Step |
Hyp |
Ref |
Expression |
1 |
|
dffun7 |
⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃* 𝑦 𝑥 𝐴 𝑦 ) ) |
2 |
|
moeu |
⊢ ( ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ( ∃ 𝑦 𝑥 𝐴 𝑦 → ∃! 𝑦 𝑥 𝐴 𝑦 ) ) |
3 |
|
vex |
⊢ 𝑥 ∈ V |
4 |
3
|
eldm |
⊢ ( 𝑥 ∈ dom 𝐴 ↔ ∃ 𝑦 𝑥 𝐴 𝑦 ) |
5 |
|
pm5.5 |
⊢ ( ∃ 𝑦 𝑥 𝐴 𝑦 → ( ( ∃ 𝑦 𝑥 𝐴 𝑦 → ∃! 𝑦 𝑥 𝐴 𝑦 ) ↔ ∃! 𝑦 𝑥 𝐴 𝑦 ) ) |
6 |
4 5
|
sylbi |
⊢ ( 𝑥 ∈ dom 𝐴 → ( ( ∃ 𝑦 𝑥 𝐴 𝑦 → ∃! 𝑦 𝑥 𝐴 𝑦 ) ↔ ∃! 𝑦 𝑥 𝐴 𝑦 ) ) |
7 |
2 6
|
bitrid |
⊢ ( 𝑥 ∈ dom 𝐴 → ( ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ∃! 𝑦 𝑥 𝐴 𝑦 ) ) |
8 |
7
|
ralbiia |
⊢ ( ∀ 𝑥 ∈ dom 𝐴 ∃* 𝑦 𝑥 𝐴 𝑦 ↔ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 ) |
9 |
8
|
anbi2i |
⊢ ( ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃* 𝑦 𝑥 𝐴 𝑦 ) ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 ) ) |
10 |
1 9
|
bitri |
⊢ ( Fun 𝐴 ↔ ( Rel 𝐴 ∧ ∀ 𝑥 ∈ dom 𝐴 ∃! 𝑦 𝑥 𝐴 𝑦 ) ) |