Step |
Hyp |
Ref |
Expression |
1 |
|
imadisj |
⊢ ( ( 𝐹 “ 𝐵 ) = ∅ ↔ ( dom 𝐹 ∩ 𝐵 ) = ∅ ) |
2 |
|
incom |
⊢ ( dom 𝐹 ∩ 𝐵 ) = ( 𝐵 ∩ dom 𝐹 ) |
3 |
|
fndm |
⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 ) |
4 |
3
|
sseq2d |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴 ) ) |
5 |
4
|
biimpar |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ dom 𝐹 ) |
6 |
|
df-ss |
⊢ ( 𝐵 ⊆ dom 𝐹 ↔ ( 𝐵 ∩ dom 𝐹 ) = 𝐵 ) |
7 |
5 6
|
sylib |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐵 ∩ dom 𝐹 ) = 𝐵 ) |
8 |
2 7
|
eqtrid |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( dom 𝐹 ∩ 𝐵 ) = 𝐵 ) |
9 |
8
|
eqeq1d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ( dom 𝐹 ∩ 𝐵 ) = ∅ ↔ 𝐵 = ∅ ) ) |
10 |
1 9
|
syl5bb |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝐹 “ 𝐵 ) = ∅ ↔ 𝐵 = ∅ ) ) |