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Metamath Proof Explorer

Theorem ind1

Description: Value of the indicator function where it is 1 . (Contributed by Thierry Arnoux, 14-Aug-2017)

Ref Expression
Assertion ind1 ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = 1 )

Proof

Step Hyp Ref Expression
1 simp2 ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → 𝐴 ⊆ 𝑂 )
2 simp3 ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ 𝐴 )
3 1 2 sseldd ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → 𝑋 ∈ 𝑂 )
4 indfval ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )
5 3 4 syld3an3 ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = if ( 𝑋 ∈ 𝐴 , 1 , 0 ) )
6 iftrue ⊢ ( 𝑋 ∈ 𝐴 → if ( 𝑋 ∈ 𝐴 , 1 , 0 ) = 1 )
7 6 3ad2ant3 ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → if ( 𝑋 ∈ 𝐴 , 1 , 0 ) = 1 )
8 5 7 eqtrd ⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴 ) → ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑋 ) = 1 )