Theorem unirnffid

Description: The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	unirnffid.1	⊢ ( 𝜑 → 𝐹 : 𝑇 ⟶ Fin )
	unirnffid.2	⊢ ( 𝜑 → 𝑇 ∈ Fin )
Assertion	unirnffid	⊢ ( 𝜑 → ∪ ran 𝐹 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		unirnffid.1	⊢ ( 𝜑 → 𝐹 : 𝑇 ⟶ Fin )
2		unirnffid.2	⊢ ( 𝜑 → 𝑇 ∈ Fin )
3	1	ffnd	⊢ ( 𝜑 → 𝐹 Fn 𝑇 )
4		fnfi	⊢ ( ( 𝐹 Fn 𝑇 ∧ 𝑇 ∈ Fin ) → 𝐹 ∈ Fin )
5	3 2 4	syl2anc	⊢ ( 𝜑 → 𝐹 ∈ Fin )
6		rnfi	⊢ ( 𝐹 ∈ Fin → ran 𝐹 ∈ Fin )
7	5 6	syl	⊢ ( 𝜑 → ran 𝐹 ∈ Fin )
8	1	frnd	⊢ ( 𝜑 → ran 𝐹 ⊆ Fin )
9		unifi	⊢ ( ( ran 𝐹 ∈ Fin ∧ ran 𝐹 ⊆ Fin ) → ∪ ran 𝐹 ∈ Fin )
10	7 8 9	syl2anc	⊢ ( 𝜑 → ∪ ran 𝐹 ∈ Fin )