Step |
Hyp |
Ref |
Expression |
1 |
|
f1ocpbl.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –1-1-onto→ 𝑋 ) |
2 |
|
f1of1 |
⊢ ( 𝐹 : 𝑉 –1-1-onto→ 𝑋 → 𝐹 : 𝑉 –1-1→ 𝑋 ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝑉 –1-1→ 𝑋 ) |
4 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐹 : 𝑉 –1-1→ 𝑋 ) |
5 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐵 ∈ 𝑉 ) |
6 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐶 ∈ 𝑉 ) |
7 |
|
f1fveq |
⊢ ( ( 𝐹 : 𝑉 –1-1→ 𝑋 ∧ ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐶 ) ↔ 𝐵 = 𝐶 ) ) |
8 |
4 5 6 7
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐶 ) ↔ 𝐵 = 𝐶 ) ) |
9 |
|
oveq2 |
⊢ ( 𝐵 = 𝐶 → ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝐵 = 𝐶 → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐴 + 𝐶 ) ) ) |
11 |
8 10
|
syl6bi |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝐵 ) = ( 𝐹 ‘ 𝐶 ) → ( 𝐹 ‘ ( 𝐴 + 𝐵 ) ) = ( 𝐹 ‘ ( 𝐴 + 𝐶 ) ) ) ) |