Step |
Hyp |
Ref |
Expression |
1 |
|
simpr1 |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ) |
2 |
|
dfiin2g |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) |
3 |
1 2
|
syl |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) |
4 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
5 |
4
|
rnmpt |
⊢ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } |
6 |
5
|
inteqi |
⊢ ∩ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ∩ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } |
7 |
3 6
|
eqtr4di |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
8 |
4
|
fmpt |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝐶 ) |
9 |
8
|
3anbi1i |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) |
10 |
|
intrnfi |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∩ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( fi ‘ 𝐶 ) ) |
11 |
9 10
|
sylan2b |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∩ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( fi ‘ 𝐶 ) ) |
12 |
7 11
|
eqeltrd |
⊢ ( ( 𝐶 ∈ 𝑉 ∧ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin ) ) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ ( fi ‘ 𝐶 ) ) |