Step |
Hyp |
Ref |
Expression |
1 |
|
ind1a |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑥 ∈ 𝑂 ) → ( ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑥 ) = 1 ↔ 𝑥 ∈ 𝐴 ) ) |
2 |
1
|
3expia |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( 𝑥 ∈ 𝑂 → ( ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑥 ) = 1 ↔ 𝑥 ∈ 𝐴 ) ) ) |
3 |
2
|
pm5.32d |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( ( 𝑥 ∈ 𝑂 ∧ ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑥 ) = 1 ) ↔ ( 𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴 ) ) ) |
4 |
|
indf |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) : 𝑂 ⟶ { 0 , 1 } ) |
5 |
|
ffn |
⊢ ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) : 𝑂 ⟶ { 0 , 1 } → ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) Fn 𝑂 ) |
6 |
|
fniniseg |
⊢ ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) Fn 𝑂 → ( 𝑥 ∈ ( ◡ ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) “ { 1 } ) ↔ ( 𝑥 ∈ 𝑂 ∧ ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑥 ) = 1 ) ) ) |
7 |
4 5 6
|
3syl |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( 𝑥 ∈ ( ◡ ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) “ { 1 } ) ↔ ( 𝑥 ∈ 𝑂 ∧ ( ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) ‘ 𝑥 ) = 1 ) ) ) |
8 |
|
ssel |
⊢ ( 𝐴 ⊆ 𝑂 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑂 ) ) |
9 |
8
|
pm4.71rd |
⊢ ( 𝐴 ⊆ 𝑂 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴 ) ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴 ) ) ) |
11 |
3 7 10
|
3bitr4d |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( 𝑥 ∈ ( ◡ ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) “ { 1 } ) ↔ 𝑥 ∈ 𝐴 ) ) |
12 |
11
|
eqrdv |
⊢ ( ( 𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ) → ( ◡ ( ( 𝟭 ‘ 𝑂 ) ‘ 𝐴 ) “ { 1 } ) = 𝐴 ) |