Metamath Proof Explorer


Theorem abrexex

Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg . See also abrexex2 . (Contributed by NM, 16-Oct-2003) (Proof shortened by Mario Carneiro, 31-Aug-2015)

Ref Expression
Hypothesis abrexex.1 𝐴 ∈ V
Assertion abrexex { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } ∈ V

Proof

Step Hyp Ref Expression
1 abrexex.1 𝐴 ∈ V
2 abrexexg ( 𝐴 ∈ V → { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } ∈ V )
3 1 2 ax-mp { 𝑦 ∣ ∃ 𝑥𝐴 𝑦 = 𝐵 } ∈ V