Step |
Hyp |
Ref |
Expression |
1 |
|
f1resrcmplf1d.1 |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 ) |
2 |
|
f1resrcmplf1d.2 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
3 |
|
f1resrcmplf1d.3 |
⊢ ( 𝜑 → ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 ) |
4 |
|
f1resrcmplf1d.4 |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 ∖ 𝐶 ) ) : ( 𝐴 ∖ 𝐶 ) –1-1→ 𝐵 ) |
5 |
|
f1resrcmplf1d.5 |
⊢ ( 𝜑 → ( ( 𝐹 “ 𝐶 ) ∩ ( 𝐹 “ ( 𝐴 ∖ 𝐶 ) ) ) = ∅ ) |
6 |
|
f1resveqaeq |
⊢ ( ( ( 𝐹 ↾ 𝐶 ) : 𝐶 –1-1→ 𝐵 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
7 |
3 6
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
8 |
7
|
ex |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
9 |
|
difssd |
⊢ ( 𝜑 → ( 𝐴 ∖ 𝐶 ) ⊆ 𝐴 ) |
10 |
1 9 2 5
|
f1resrcmplf1dlem |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
11 |
|
incom |
⊢ ( ( 𝐹 “ 𝐶 ) ∩ ( 𝐹 “ ( 𝐴 ∖ 𝐶 ) ) ) = ( ( 𝐹 “ ( 𝐴 ∖ 𝐶 ) ) ∩ ( 𝐹 “ 𝐶 ) ) |
12 |
11 5
|
eqtr3id |
⊢ ( 𝜑 → ( ( 𝐹 “ ( 𝐴 ∖ 𝐶 ) ) ∩ ( 𝐹 “ 𝐶 ) ) = ∅ ) |
13 |
9 1 2 12
|
f1resrcmplf1dlem |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ∧ 𝑦 ∈ 𝐶 ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
14 |
|
f1resveqaeq |
⊢ ( ( ( 𝐹 ↾ ( 𝐴 ∖ 𝐶 ) ) : ( 𝐴 ∖ 𝐶 ) –1-1→ 𝐵 ∧ ( 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ∧ 𝑦 ∈ ( 𝐴 ∖ 𝐶 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
15 |
4 14
|
sylan |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ∧ 𝑦 ∈ ( 𝐴 ∖ 𝐶 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
16 |
15
|
ex |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 ∖ 𝐶 ) ∧ 𝑦 ∈ ( 𝐴 ∖ 𝐶 ) ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
17 |
8 10 13 16
|
prsrcmpltd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
18 |
17
|
ralrimivv |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) |
19 |
|
dff13 |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
20 |
2 18 19
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 : 𝐴 –1-1→ 𝐵 ) |