Theorem abrexfi

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

	Ref	Expression
Assertion	abrexfi	⊢ ( 𝐴 ∈ Fin → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ Fin )

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		mptfi	⊢ ( 𝐴 ∈ Fin → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin )
4		rnfi	⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin )
5	3 4	syl	⊢ ( 𝐴 ∈ Fin → ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ Fin )
6	2 5	eqeltrrid	⊢ ( 𝐴 ∈ Fin → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ∈ Fin )