Theorem rabfi

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

	Ref	Expression
Assertion	rabfi	$$⊢A\in \mathrm{Fin}\to \left\{x\in A|\phi \right\}\in \mathrm{Fin}$$

Proof

Step	Hyp	Ref	Expression
1		dfrab3	$$⊢\left\{x\in A|\phi \right\}=A\cap \left\{x|\phi \right\}$$
2		infi	$$⊢A\in \mathrm{Fin}\to A\cap \left\{x|\phi \right\}\in \mathrm{Fin}$$
3	1 2	eqeltrid	$$⊢A\in \mathrm{Fin}\to \left\{x\in A|\phi \right\}\in \mathrm{Fin}$$