Theorem abrexfi

Description: An image set from a finite set is finite. (Contributed by Mario Carneiro, 13-Feb-2014)

	Ref	Expression
Assertion	abrexfi	|- ( A e. Fin -> { y | E. x e. A y = B } e. Fin )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
2	1	rnmpt	|- ran ( x e. A |-> B ) = { y | E. x e. A y = B }
3		mptfi	|- ( A e. Fin -> ( x e. A |-> B ) e. Fin )
4		rnfi	|- ( ( x e. A |-> B ) e. Fin -> ran ( x e. A |-> B ) e. Fin )
5	3 4	syl	|- ( A e. Fin -> ran ( x e. A |-> B ) e. Fin )
6	2 5	eqeltrrid	|- ( A e. Fin -> { y | E. x e. A y = B } e. Fin )