Theorem rabfi

Description: A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017)

	Ref	Expression
Assertion	rabfi	⊢ ( 𝐴 ∈ Fin → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		dfrab3	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = ( 𝐴 ∩ { 𝑥 ∣ 𝜑 } )
2		infi	⊢ ( 𝐴 ∈ Fin → ( 𝐴 ∩ { 𝑥 ∣ 𝜑 } ) ∈ Fin )
3	1 2	eqeltrid	⊢ ( 𝐴 ∈ Fin → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ Fin )