Metamath Proof Explorer
Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007)
|
|
Ref |
Expression |
|
Hypotheses |
fabex.1 |
⊢ 𝐴 ∈ V |
|
|
fabex.2 |
⊢ 𝐵 ∈ V |
|
|
fabex.3 |
⊢ 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) } |
|
Assertion |
fabex |
⊢ 𝐹 ∈ V |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
fabex.1 |
⊢ 𝐴 ∈ V |
2 |
|
fabex.2 |
⊢ 𝐵 ∈ V |
3 |
|
fabex.3 |
⊢ 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) } |
4 |
3
|
fabexg |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 𝐹 ∈ V ) |
5 |
1 2 4
|
mp2an |
⊢ 𝐹 ∈ V |