Metamath Proof Explorer
Description: The function value of an endofunction. (Contributed by AV, 27-Jan-2024)
|
|
Ref |
Expression |
|
Hypotheses |
efmndbas.g |
⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) |
|
|
efmndbas.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
Assertion |
efmndfv |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
efmndbas.g |
⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) |
2 |
|
efmndbas.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
3 |
1 2
|
efmndbasf |
⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐴 ) |
4 |
3
|
ffvelrnda |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 ) |