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 ) |
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 ) |