Step |
Hyp |
Ref |
Expression |
1 |
|
df-fn |
⊢ ( ( 𝐹 ↾ 𝐵 ) Fn 𝐵 ↔ ( Fun ( 𝐹 ↾ 𝐵 ) ∧ dom ( 𝐹 ↾ 𝐵 ) = 𝐵 ) ) |
2 |
|
fnfun |
⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 ) |
3 |
2
|
funresd |
⊢ ( 𝐹 Fn 𝐴 → Fun ( 𝐹 ↾ 𝐵 ) ) |
4 |
3
|
biantrurd |
⊢ ( 𝐹 Fn 𝐴 → ( dom ( 𝐹 ↾ 𝐵 ) = 𝐵 ↔ ( Fun ( 𝐹 ↾ 𝐵 ) ∧ dom ( 𝐹 ↾ 𝐵 ) = 𝐵 ) ) ) |
5 |
|
ssdmres |
⊢ ( 𝐵 ⊆ dom 𝐹 ↔ dom ( 𝐹 ↾ 𝐵 ) = 𝐵 ) |
6 |
|
fndm |
⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 ) |
7 |
6
|
sseq2d |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴 ) ) |
8 |
5 7
|
bitr3id |
⊢ ( 𝐹 Fn 𝐴 → ( dom ( 𝐹 ↾ 𝐵 ) = 𝐵 ↔ 𝐵 ⊆ 𝐴 ) ) |
9 |
4 8
|
bitr3d |
⊢ ( 𝐹 Fn 𝐴 → ( ( Fun ( 𝐹 ↾ 𝐵 ) ∧ dom ( 𝐹 ↾ 𝐵 ) = 𝐵 ) ↔ 𝐵 ⊆ 𝐴 ) ) |
10 |
1 9
|
bitrid |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐹 ↾ 𝐵 ) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴 ) ) |