Metamath Proof Explorer
Description: A singleton index picks out an instance of an indexed union's argument.
(Contributed by Glauco Siliprandi, 17-Aug-2020)
|
|
Ref |
Expression |
|
Hypotheses |
iunxsnf.1 |
⊢ Ⅎ 𝑥 𝐶 |
|
|
iunxsnf.2 |
⊢ 𝐴 ∈ V |
|
|
iunxsnf.3 |
⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) |
|
Assertion |
iunxsnf |
⊢ ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
iunxsnf.1 |
⊢ Ⅎ 𝑥 𝐶 |
2 |
|
iunxsnf.2 |
⊢ 𝐴 ∈ V |
3 |
|
iunxsnf.3 |
⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 ) |
4 |
1 3
|
iunxsngf |
⊢ ( 𝐴 ∈ V → ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 ) |
5 |
2 4
|
ax-mp |
⊢ ∪ 𝑥 ∈ { 𝐴 } 𝐵 = 𝐶 |