Theorem abrexct

Description: An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016) (Moved to the main part of set.mm by Vincent Gonzalez, 19-Aug-2026.)

	Ref	Expression
Assertion	abrexct	⊢ ( 𝐴 ≼ ω → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ≼ ω )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2	1	rnmpt	⊢ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }
3		1stcrestlem	⊢ ( 𝐴 ≼ ω → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ≼ ω )
4	2 3	eqbrtrrid	⊢ ( 𝐴 ≼ ω → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ≼ ω )