Step |
Hyp |
Ref |
Expression |
1 |
|
dfiin2g |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) |
2 |
1
|
adantl |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) |
3 |
|
elisset |
⊢ ( 𝐵 ∈ 𝐶 → ∃ 𝑦 𝑦 = 𝐵 ) |
4 |
3
|
rgenw |
⊢ ∀ 𝑥 ∈ 𝐴 ( 𝐵 ∈ 𝐶 → ∃ 𝑦 𝑦 = 𝐵 ) |
5 |
|
r19.2z |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐵 ∈ 𝐶 → ∃ 𝑦 𝑦 = 𝐵 ) ) → ∃ 𝑥 ∈ 𝐴 ( 𝐵 ∈ 𝐶 → ∃ 𝑦 𝑦 = 𝐵 ) ) |
6 |
4 5
|
mpan2 |
⊢ ( 𝐴 ≠ ∅ → ∃ 𝑥 ∈ 𝐴 ( 𝐵 ∈ 𝐶 → ∃ 𝑦 𝑦 = 𝐵 ) ) |
7 |
|
r19.35 |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝐵 ∈ 𝐶 → ∃ 𝑦 𝑦 = 𝐵 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 = 𝐵 ) ) |
8 |
6 7
|
sylib |
⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 = 𝐵 ) ) |
9 |
8
|
imp |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 = 𝐵 ) |
10 |
|
rexcom4 |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 𝑦 = 𝐵 ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
11 |
9 10
|
sylib |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) → ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
12 |
|
abn0 |
⊢ ( { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ≠ ∅ ↔ ∃ 𝑦 ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ) |
13 |
11 12
|
sylibr |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ≠ ∅ ) |
14 |
|
intex |
⊢ ( { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ≠ ∅ ↔ ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ V ) |
15 |
13 14
|
sylib |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) → ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ V ) |
16 |
2 15
|
eqeltrd |
⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V ) |