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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 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ V

Proof

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