Step |
Hyp |
Ref |
Expression |
1 |
|
eqresfnbd.g |
⊢ ( 𝜑 → 𝐹 Fn 𝐵 ) |
2 |
|
eqresfnbd.1 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
3 |
1 2
|
fnssresd |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐴 ) Fn 𝐴 ) |
4 |
|
resss |
⊢ ( 𝐹 ↾ 𝐴 ) ⊆ 𝐹 |
5 |
3 4
|
jctir |
⊢ ( 𝜑 → ( ( 𝐹 ↾ 𝐴 ) Fn 𝐴 ∧ ( 𝐹 ↾ 𝐴 ) ⊆ 𝐹 ) ) |
6 |
|
fneq1 |
⊢ ( 𝑅 = ( 𝐹 ↾ 𝐴 ) → ( 𝑅 Fn 𝐴 ↔ ( 𝐹 ↾ 𝐴 ) Fn 𝐴 ) ) |
7 |
|
sseq1 |
⊢ ( 𝑅 = ( 𝐹 ↾ 𝐴 ) → ( 𝑅 ⊆ 𝐹 ↔ ( 𝐹 ↾ 𝐴 ) ⊆ 𝐹 ) ) |
8 |
6 7
|
anbi12d |
⊢ ( 𝑅 = ( 𝐹 ↾ 𝐴 ) → ( ( 𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹 ) ↔ ( ( 𝐹 ↾ 𝐴 ) Fn 𝐴 ∧ ( 𝐹 ↾ 𝐴 ) ⊆ 𝐹 ) ) ) |
9 |
5 8
|
syl5ibrcom |
⊢ ( 𝜑 → ( 𝑅 = ( 𝐹 ↾ 𝐴 ) → ( 𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹 ) ) ) |
10 |
1
|
fnfund |
⊢ ( 𝜑 → Fun 𝐹 ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑅 Fn 𝐴 ) → Fun 𝐹 ) |
12 |
|
funssres |
⊢ ( ( Fun 𝐹 ∧ 𝑅 ⊆ 𝐹 ) → ( 𝐹 ↾ dom 𝑅 ) = 𝑅 ) |
13 |
12
|
eqcomd |
⊢ ( ( Fun 𝐹 ∧ 𝑅 ⊆ 𝐹 ) → 𝑅 = ( 𝐹 ↾ dom 𝑅 ) ) |
14 |
|
fndm |
⊢ ( 𝑅 Fn 𝐴 → dom 𝑅 = 𝐴 ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑅 Fn 𝐴 ) → dom 𝑅 = 𝐴 ) |
16 |
15
|
reseq2d |
⊢ ( ( 𝜑 ∧ 𝑅 Fn 𝐴 ) → ( 𝐹 ↾ dom 𝑅 ) = ( 𝐹 ↾ 𝐴 ) ) |
17 |
16
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑅 Fn 𝐴 ) → ( 𝑅 = ( 𝐹 ↾ dom 𝑅 ) ↔ 𝑅 = ( 𝐹 ↾ 𝐴 ) ) ) |
18 |
13 17
|
imbitrid |
⊢ ( ( 𝜑 ∧ 𝑅 Fn 𝐴 ) → ( ( Fun 𝐹 ∧ 𝑅 ⊆ 𝐹 ) → 𝑅 = ( 𝐹 ↾ 𝐴 ) ) ) |
19 |
11 18
|
mpand |
⊢ ( ( 𝜑 ∧ 𝑅 Fn 𝐴 ) → ( 𝑅 ⊆ 𝐹 → 𝑅 = ( 𝐹 ↾ 𝐴 ) ) ) |
20 |
19
|
expimpd |
⊢ ( 𝜑 → ( ( 𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹 ) → 𝑅 = ( 𝐹 ↾ 𝐴 ) ) ) |
21 |
9 20
|
impbid |
⊢ ( 𝜑 → ( 𝑅 = ( 𝐹 ↾ 𝐴 ) ↔ ( 𝑅 Fn 𝐴 ∧ 𝑅 ⊆ 𝐹 ) ) ) |