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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 ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = 0 )

Proof

Step Hyp Ref Expression
1 eldifi ⊢ ( 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) → 𝑋 ∈ 𝑂 )
2 indfval ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )
3 1 2 syl3an3 ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )
4 eldifn ⊢ ( 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) → ¬ 𝑋 ∈ 𝐴 )
5 4 3ad2ant3 ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) ) → ¬ 𝑋 ∈ 𝐴 )
6 5 iffalsed ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) ) → if ( 𝑋 ∈ 𝐴 , 1 , 0 ) = 0 )
7 3 6 eqtrd ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ ( 𝑂 ∖ 𝐴 ) ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = 0 )