Step |
Hyp |
Ref |
Expression |
1 |
|
tskmval |
⊢ ( 𝐴 ∈ 𝑉 → ( tarskiMap ‘ 𝐴 ) = ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ) |
2 |
1
|
eleq2d |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ↔ 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ) ) |
3 |
|
elex |
⊢ ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } → 𝐵 ∈ V ) |
4 |
3
|
a1i |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } → 𝐵 ∈ V ) ) |
5 |
|
tskmid |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) ) |
6 |
|
tskmcl |
⊢ ( tarskiMap ‘ 𝐴 ) ∈ Tarski |
7 |
|
eleq2 |
⊢ ( 𝑥 = ( tarskiMap ‘ 𝐴 ) → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) ) ) |
8 |
|
eleq2 |
⊢ ( 𝑥 = ( tarskiMap ‘ 𝐴 ) → ( 𝐵 ∈ 𝑥 ↔ 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) |
9 |
7 8
|
imbi12d |
⊢ ( 𝑥 = ( tarskiMap ‘ 𝐴 ) → ( ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ↔ ( 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) ) |
10 |
9
|
rspcv |
⊢ ( ( tarskiMap ‘ 𝐴 ) ∈ Tarski → ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → ( 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) ) |
11 |
6 10
|
ax-mp |
⊢ ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → ( 𝐴 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) |
12 |
5 11
|
syl5com |
⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ) ) |
13 |
|
elex |
⊢ ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) → 𝐵 ∈ V ) |
14 |
12 13
|
syl6 |
⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) → 𝐵 ∈ V ) ) |
15 |
|
elintrabg |
⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) ) |
16 |
15
|
a1i |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ V → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) ) ) |
17 |
4 14 16
|
pm5.21ndd |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ∩ { 𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥 } ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) ) |
18 |
2 17
|
bitrd |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐵 ∈ ( tarskiMap ‘ 𝐴 ) ↔ ∀ 𝑥 ∈ Tarski ( 𝐴 ∈ 𝑥 → 𝐵 ∈ 𝑥 ) ) ) |