Theorem rnmptfi

Description: The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypothesis	rnmptfi.a	⊢ 𝐴 = ( 𝑥 ∈ 𝐵 ↦ 𝐶 )
Assertion	rnmptfi	⊢ ( 𝐵 ∈ Fin → ran 𝐴 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		rnmptfi.a	⊢ 𝐴 = ( 𝑥 ∈ 𝐵 ↦ 𝐶 )
2		mptfi	⊢ ( 𝐵 ∈ Fin → ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ Fin )
3	1 2	eqeltrid	⊢ ( 𝐵 ∈ Fin → 𝐴 ∈ Fin )
4		rnfi	⊢ ( 𝐴 ∈ Fin → ran 𝐴 ∈ Fin )
5	3 4	syl	⊢ ( 𝐵 ∈ Fin → ran 𝐴 ∈ Fin )