Metamath Proof Explorer
Description: A function's value belongs to its range. (Contributed by Glauco
Siliprandi, 2-Jan-2022)
|
|
Ref |
Expression |
|
Hypotheses |
fnfvelrnd.1 |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
|
|
fnfvelrnd.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
|
Assertion |
fnfvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
fnfvelrnd.1 |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
2 |
|
fnfvelrnd.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
3 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 ) |