Metamath Proof Explorer
Description: The monoid of endofunctions on a finite set A is finite.
(Contributed by AV, 27-Jan-2024)
|
|
Ref |
Expression |
|
Hypotheses |
efmndbas.g |
⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) |
|
|
efmndbas.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
Assertion |
efmndbasfi |
⊢ ( 𝐴 ∈ Fin → 𝐵 ∈ Fin ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
efmndbas.g |
⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) |
2 |
|
efmndbas.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
3 |
1 2
|
efmndbas |
⊢ 𝐵 = ( 𝐴 ↑m 𝐴 ) |
4 |
|
mapfi |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐴 ∈ Fin ) → ( 𝐴 ↑m 𝐴 ) ∈ Fin ) |
5 |
4
|
anidms |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 ↑m 𝐴 ) ∈ Fin ) |
6 |
3 5
|
eqeltrid |
⊢ ( 𝐴 ∈ Fin → 𝐵 ∈ Fin ) |