Step |
Hyp |
Ref |
Expression |
1 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ) |
2 |
1
|
anbi1i |
⊢ ( ( 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) |
3 |
|
r19.41v |
⊢ ( ∃ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ↔ ( ∃ 𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) |
4 |
2 3
|
bitr4i |
⊢ ( ( 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑 ) ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) |
5 |
4
|
abbii |
⊢ { 𝑥 ∣ ( 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑 ) } = { 𝑥 ∣ ∃ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } |
6 |
|
df-rab |
⊢ { 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝜑 ) } |
7 |
|
iunab |
⊢ ∪ 𝑦 ∈ 𝐴 { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } = { 𝑥 ∣ ∃ 𝑦 ∈ 𝐴 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } |
8 |
5 6 7
|
3eqtr4i |
⊢ { 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑 } = ∪ 𝑦 ∈ 𝐴 { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } |
9 |
|
df-rab |
⊢ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } |
10 |
9
|
a1i |
⊢ ( 𝑦 ∈ 𝐴 → { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } ) |
11 |
10
|
iuneq2i |
⊢ ∪ 𝑦 ∈ 𝐴 { 𝑥 ∈ 𝐵 ∣ 𝜑 } = ∪ 𝑦 ∈ 𝐴 { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) } |
12 |
8 11
|
eqtr4i |
⊢ { 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∣ 𝜑 } = ∪ 𝑦 ∈ 𝐴 { 𝑥 ∈ 𝐵 ∣ 𝜑 } |