Metamath Proof Explorer


Theorem ind0

Description: Value of the indicator function where it is 0 . (Contributed by Thierry Arnoux, 14-Aug-2017)

Ref Expression
Assertion ind0 ( ( 𝑂𝑉𝐴𝑂𝑋 ∈ ( 𝑂𝐴 ) ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = 0 )

Proof

Step Hyp Ref Expression
1 eldifi ( 𝑋 ∈ ( 𝑂𝐴 ) → 𝑋𝑂 )
2 indfval ( ( 𝑂𝑉𝐴𝑂𝑋𝑂 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋𝐴 , 1 , 0 ) )
3 1 2 syl3an3 ( ( 𝑂𝑉𝐴𝑂𝑋 ∈ ( 𝑂𝐴 ) ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋𝐴 , 1 , 0 ) )
4 eldifn ( 𝑋 ∈ ( 𝑂𝐴 ) → ¬ 𝑋𝐴 )
5 4 3ad2ant3 ( ( 𝑂𝑉𝐴𝑂𝑋 ∈ ( 𝑂𝐴 ) ) → ¬ 𝑋𝐴 )
6 5 iffalsed ( ( 𝑂𝑉𝐴𝑂𝑋 ∈ ( 𝑂𝐴 ) ) → if ( 𝑋𝐴 , 1 , 0 ) = 0 )
7 3 6 eqtrd ( ( 𝑂𝑉𝐴𝑂𝑋 ∈ ( 𝑂𝐴 ) ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = 0 )