Metamath Proof Explorer
Description: Alternate definition of indexed union when B is a set. (Contributed by Mario Carneiro, 31-Aug-2015)
|
|
Ref |
Expression |
|
Hypothesis |
dfiun3.1 |
⊢ 𝐵 ∈ V |
|
Assertion |
dfiun3 |
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
dfiun3.1 |
⊢ 𝐵 ∈ V |
2 |
|
dfiun3g |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ V → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
3 |
1
|
a1i |
⊢ ( 𝑥 ∈ 𝐴 → 𝐵 ∈ V ) |
4 |
2 3
|
mprg |
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |