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 |
|
mapvalg |
⊢ ( ( 𝐴 ∈ V ∧ 𝐴 ∈ V ) → ( 𝐴 ↑m 𝐴 ) = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } ) |
5 |
4
|
anidms |
⊢ ( 𝐴 ∈ V → ( 𝐴 ↑m 𝐴 ) = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } ) |
6 |
3 5
|
eqtrid |
⊢ ( 𝐴 ∈ V → 𝐵 = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } ) |
7 |
|
base0 |
⊢ ∅ = ( Base ‘ ∅ ) |
8 |
7
|
eqcomi |
⊢ ( Base ‘ ∅ ) = ∅ |
9 |
|
fvprc |
⊢ ( ¬ 𝐴 ∈ V → ( EndoFMnd ‘ 𝐴 ) = ∅ ) |
10 |
1 9
|
eqtrid |
⊢ ( ¬ 𝐴 ∈ V → 𝐺 = ∅ ) |
11 |
10
|
fveq2d |
⊢ ( ¬ 𝐴 ∈ V → ( Base ‘ 𝐺 ) = ( Base ‘ ∅ ) ) |
12 |
2 11
|
eqtrid |
⊢ ( ¬ 𝐴 ∈ V → 𝐵 = ( Base ‘ ∅ ) ) |
13 |
|
mapprc |
⊢ ( ¬ 𝐴 ∈ V → { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } = ∅ ) |
14 |
8 12 13
|
3eqtr4a |
⊢ ( ¬ 𝐴 ∈ V → 𝐵 = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } ) |
15 |
6 14
|
pm2.61i |
⊢ 𝐵 = { 𝑓 ∣ 𝑓 : 𝐴 ⟶ 𝐴 } |