Step |
Hyp |
Ref |
Expression |
1 |
|
funfn |
⊢ ( Fun 𝐹 ↔ 𝐹 Fn dom 𝐹 ) |
2 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹 ) ) |
3 |
2
|
biancomi |
⊢ ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ↔ ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ) |
4 |
3
|
anbi1i |
⊢ ( ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ) |
5 |
|
fvres |
⊢ ( 𝑥 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
6 |
5
|
eleq1d |
⊢ ( 𝑥 ∈ 𝐵 → ( ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ↔ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) → ( ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ↔ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ) |
8 |
7
|
pm5.32i |
⊢ ( ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ) |
9 |
4 8
|
bitri |
⊢ ( ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ) |
10 |
9
|
a1i |
⊢ ( 𝐹 Fn dom 𝐹 → ( ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ) ) |
11 |
|
an32 |
⊢ ( ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐵 ) ) |
12 |
10 11
|
bitrdi |
⊢ ( 𝐹 Fn dom 𝐹 → ( ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐵 ) ) ) |
13 |
|
fnfun |
⊢ ( 𝐹 Fn dom 𝐹 → Fun 𝐹 ) |
14 |
13
|
funresd |
⊢ ( 𝐹 Fn dom 𝐹 → Fun ( 𝐹 ↾ 𝐵 ) ) |
15 |
|
dmres |
⊢ dom ( 𝐹 ↾ 𝐵 ) = ( 𝐵 ∩ dom 𝐹 ) |
16 |
|
df-fn |
⊢ ( ( 𝐹 ↾ 𝐵 ) Fn ( 𝐵 ∩ dom 𝐹 ) ↔ ( Fun ( 𝐹 ↾ 𝐵 ) ∧ dom ( 𝐹 ↾ 𝐵 ) = ( 𝐵 ∩ dom 𝐹 ) ) ) |
17 |
14 15 16
|
sylanblrc |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝐹 ↾ 𝐵 ) Fn ( 𝐵 ∩ dom 𝐹 ) ) |
18 |
|
elpreima |
⊢ ( ( 𝐹 ↾ 𝐵 ) Fn ( 𝐵 ∩ dom 𝐹 ) → ( 𝑥 ∈ ( ◡ ( 𝐹 ↾ 𝐵 ) “ 𝐴 ) ↔ ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ) ) |
19 |
17 18
|
syl |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝑥 ∈ ( ◡ ( 𝐹 ↾ 𝐵 ) “ 𝐴 ) ↔ ( 𝑥 ∈ ( 𝐵 ∩ dom 𝐹 ) ∧ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝑥 ) ∈ 𝐴 ) ) ) |
20 |
|
elin |
⊢ ( 𝑥 ∈ ( ( ◡ 𝐹 “ 𝐴 ) ∩ 𝐵 ) ↔ ( 𝑥 ∈ ( ◡ 𝐹 “ 𝐴 ) ∧ 𝑥 ∈ 𝐵 ) ) |
21 |
|
elpreima |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝑥 ∈ ( ◡ 𝐹 “ 𝐴 ) ↔ ( 𝑥 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ) ) |
22 |
21
|
anbi1d |
⊢ ( 𝐹 Fn dom 𝐹 → ( ( 𝑥 ∈ ( ◡ 𝐹 “ 𝐴 ) ∧ 𝑥 ∈ 𝐵 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐵 ) ) ) |
23 |
20 22
|
bitrid |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝑥 ∈ ( ( ◡ 𝐹 “ 𝐴 ) ∩ 𝐵 ) ↔ ( ( 𝑥 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐴 ) ∧ 𝑥 ∈ 𝐵 ) ) ) |
24 |
12 19 23
|
3bitr4d |
⊢ ( 𝐹 Fn dom 𝐹 → ( 𝑥 ∈ ( ◡ ( 𝐹 ↾ 𝐵 ) “ 𝐴 ) ↔ 𝑥 ∈ ( ( ◡ 𝐹 “ 𝐴 ) ∩ 𝐵 ) ) ) |
25 |
1 24
|
sylbi |
⊢ ( Fun 𝐹 → ( 𝑥 ∈ ( ◡ ( 𝐹 ↾ 𝐵 ) “ 𝐴 ) ↔ 𝑥 ∈ ( ( ◡ 𝐹 “ 𝐴 ) ∩ 𝐵 ) ) ) |
26 |
25
|
eqrdv |
⊢ ( Fun 𝐹 → ( ◡ ( 𝐹 ↾ 𝐵 ) “ 𝐴 ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ 𝐵 ) ) |