Step |
Hyp |
Ref |
Expression |
1 |
|
fnresin1 |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) Fn ( 𝐴 ∩ 𝐵 ) ) |
2 |
|
resindi |
⊢ ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝐹 ↾ 𝐴 ) ∩ ( 𝐹 ↾ 𝐵 ) ) |
3 |
|
fnresdm |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ 𝐴 ) = 𝐹 ) |
4 |
3
|
ineq1d |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐹 ↾ 𝐴 ) ∩ ( 𝐹 ↾ 𝐵 ) ) = ( 𝐹 ∩ ( 𝐹 ↾ 𝐵 ) ) ) |
5 |
|
incom |
⊢ ( ( 𝐹 ↾ 𝐵 ) ∩ 𝐹 ) = ( 𝐹 ∩ ( 𝐹 ↾ 𝐵 ) ) |
6 |
|
resss |
⊢ ( 𝐹 ↾ 𝐵 ) ⊆ 𝐹 |
7 |
|
df-ss |
⊢ ( ( 𝐹 ↾ 𝐵 ) ⊆ 𝐹 ↔ ( ( 𝐹 ↾ 𝐵 ) ∩ 𝐹 ) = ( 𝐹 ↾ 𝐵 ) ) |
8 |
6 7
|
mpbi |
⊢ ( ( 𝐹 ↾ 𝐵 ) ∩ 𝐹 ) = ( 𝐹 ↾ 𝐵 ) |
9 |
5 8
|
eqtr3i |
⊢ ( 𝐹 ∩ ( 𝐹 ↾ 𝐵 ) ) = ( 𝐹 ↾ 𝐵 ) |
10 |
4 9
|
eqtrdi |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐹 ↾ 𝐴 ) ∩ ( 𝐹 ↾ 𝐵 ) ) = ( 𝐹 ↾ 𝐵 ) ) |
11 |
2 10
|
syl5eq |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐹 ↾ 𝐵 ) ) |
12 |
11
|
fneq1d |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐹 ↾ ( 𝐴 ∩ 𝐵 ) ) Fn ( 𝐴 ∩ 𝐵 ) ↔ ( 𝐹 ↾ 𝐵 ) Fn ( 𝐴 ∩ 𝐵 ) ) ) |
13 |
1 12
|
mpbid |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ 𝐵 ) Fn ( 𝐴 ∩ 𝐵 ) ) |