Step |
Hyp |
Ref |
Expression |
1 |
|
stdbdmet.1 |
⊢ 𝐷 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 , ( 𝑥 𝐶 𝑦 ) , 𝑅 ) ) |
2 |
|
ovex |
⊢ ( 𝐴 𝐶 𝐵 ) ∈ V |
3 |
|
ifexg |
⊢ ( ( ( 𝐴 𝐶 𝐵 ) ∈ V ∧ 𝑅 ∈ 𝑉 ) → if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ∈ V ) |
4 |
2 3
|
mpan |
⊢ ( 𝑅 ∈ 𝑉 → if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ∈ V ) |
5 |
|
oveq12 |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 𝐶 𝑦 ) = ( 𝐴 𝐶 𝐵 ) ) |
6 |
5
|
breq1d |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 ↔ ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 ) ) |
7 |
6 5
|
ifbieq1d |
⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → if ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 , ( 𝑥 𝐶 𝑦 ) , 𝑅 ) = if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ) |
8 |
7 1
|
ovmpoga |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ∈ V ) → ( 𝐴 𝐷 𝐵 ) = if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ) |
9 |
4 8
|
syl3an3 |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑅 ∈ 𝑉 ) → ( 𝐴 𝐷 𝐵 ) = if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ) |
10 |
9
|
3comr |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) ) |