Theorem fimact

Description: The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018)

	Ref	Expression
Assertion	fimact	⊢ ( ( 𝐴 ≼ ω ∧ Fun 𝐹 ) → ( 𝐹 “ 𝐴 ) ≼ ω )

Proof

Step	Hyp	Ref	Expression
1		ctex	⊢ ( 𝐴 ≼ ω → 𝐴 ∈ V )
2		imadomg	⊢ ( 𝐴 ∈ V → ( Fun 𝐹 → ( 𝐹 “ 𝐴 ) ≼ 𝐴 ) )
3	2	imp	⊢ ( ( 𝐴 ∈ V ∧ Fun 𝐹 ) → ( 𝐹 “ 𝐴 ) ≼ 𝐴 )
4	1 3	sylan	⊢ ( ( 𝐴 ≼ ω ∧ Fun 𝐹 ) → ( 𝐹 “ 𝐴 ) ≼ 𝐴 )
5		simpl	⊢ ( ( 𝐴 ≼ ω ∧ Fun 𝐹 ) → 𝐴 ≼ ω )
6		domtr	⊢ ( ( ( 𝐹 “ 𝐴 ) ≼ 𝐴 ∧ 𝐴 ≼ ω ) → ( 𝐹 “ 𝐴 ) ≼ ω )
7	4 5 6	syl2anc	⊢ ( ( 𝐴 ≼ ω ∧ Fun 𝐹 ) → ( 𝐹 “ 𝐴 ) ≼ ω )