Theorem indfval

Description: Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017)

	Ref	Expression
Assertion	indfval	⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		indval	⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) = ( 𝑥 ∈ 𝑂 ↦ if ( 𝑥 ∈ 𝐴 , 1 , 0 ) ) )
2	1	3adant3	⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) = ( 𝑥 ∈ 𝑂 ↦ if ( 𝑥 ∈ 𝐴 , 1 , 0 ) ) )
3		simpr	⊢ ( ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) ∧ 𝑥 = 𝑋 ) → 𝑥 = 𝑋 )
4	3	eleq1d	⊢ ( ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) ∧ 𝑥 = 𝑋 ) → ( 𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴 ) )
5	4	ifbid	⊢ ( ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) ∧ 𝑥 = 𝑋 ) → if ( 𝑥 ∈ 𝐴 , 1 , 0 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )
6		simp3	⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → 𝑋 ∈ 𝑂 )
7		1re	⊢ 1 ∈ ℝ
8		0re	⊢ 0 ∈ ℝ
9	7 8	ifcli	⊢ if ( 𝑋 ∈ 𝐴 , 1 , 0 ) ∈ ℝ
10	9	a1i	⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → if ( 𝑋 ∈ 𝐴 , 1 , 0 ) ∈ ℝ )
11	2 5 6 10	fvmptd	⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )