Theorem stdbdmetval

Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015)

	Ref	Expression
Hypothesis	stdbdmet.1	⊢ 𝐷 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑋 ↦ if ( ( 𝑥 𝐶 𝑦 ) ≤ 𝑅 , ( 𝑥 𝐶 𝑦 ) , 𝑅 ) )
Assertion	stdbdmetval	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = if ( ( 𝐴 𝐶 𝐵 ) ≤ 𝑅 , ( 𝐴 𝐶 𝐵 ) , 𝑅 ) )

Proof

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 ) )