Step |
Hyp |
Ref |
Expression |
1 |
|
sursubmefmnd.m |
⊢ 𝑀 = ( EndoFMnd ‘ 𝐴 ) |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
|
foeq1 |
⊢ ( ℎ = 𝑥 → ( ℎ : 𝐴 –onto→ 𝐴 ↔ 𝑥 : 𝐴 –onto→ 𝐴 ) ) |
4 |
2 3
|
elab |
⊢ ( 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ↔ 𝑥 : 𝐴 –onto→ 𝐴 ) |
5 |
|
fof |
⊢ ( 𝑥 : 𝐴 –onto→ 𝐴 → 𝑥 : 𝐴 ⟶ 𝐴 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 ) |
7 |
1 6
|
elefmndbas |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ ( Base ‘ 𝑀 ) ↔ 𝑥 : 𝐴 ⟶ 𝐴 ) ) |
8 |
5 7
|
syl5ibr |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 : 𝐴 –onto→ 𝐴 → 𝑥 ∈ ( Base ‘ 𝑀 ) ) ) |
9 |
4 8
|
syl5bi |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } → 𝑥 ∈ ( Base ‘ 𝑀 ) ) ) |
10 |
9
|
ssrdv |
⊢ ( 𝐴 ∈ 𝑉 → { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ⊆ ( Base ‘ 𝑀 ) ) |
11 |
1
|
efmndid |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) = ( 0g ‘ 𝑀 ) ) |
12 |
|
resiexg |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) ∈ V ) |
13 |
|
f1oi |
⊢ ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 |
14 |
|
f1ofo |
⊢ ( ( I ↾ 𝐴 ) : 𝐴 –1-1-onto→ 𝐴 → ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴 ) |
15 |
13 14
|
mp1i |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴 ) |
16 |
|
foeq1 |
⊢ ( ℎ = ( I ↾ 𝐴 ) → ( ℎ : 𝐴 –onto→ 𝐴 ↔ ( I ↾ 𝐴 ) : 𝐴 –onto→ 𝐴 ) ) |
17 |
12 15 16
|
elabd |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ 𝐴 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) |
18 |
11 17
|
eqeltrrd |
⊢ ( 𝐴 ∈ 𝑉 → ( 0g ‘ 𝑀 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) |
19 |
|
vex |
⊢ 𝑦 ∈ V |
20 |
|
foeq1 |
⊢ ( ℎ = 𝑦 → ( ℎ : 𝐴 –onto→ 𝐴 ↔ 𝑦 : 𝐴 –onto→ 𝐴 ) ) |
21 |
19 20
|
elab |
⊢ ( 𝑦 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ↔ 𝑦 : 𝐴 –onto→ 𝐴 ) |
22 |
4 21
|
anbi12i |
⊢ ( ( 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∧ 𝑦 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) ↔ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) |
23 |
|
foco |
⊢ ( ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) → ( 𝑥 ∘ 𝑦 ) : 𝐴 –onto→ 𝐴 ) |
24 |
23
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) → ( 𝑥 ∘ 𝑦 ) : 𝐴 –onto→ 𝐴 ) |
25 |
|
fof |
⊢ ( 𝑦 : 𝐴 –onto→ 𝐴 → 𝑦 : 𝐴 ⟶ 𝐴 ) |
26 |
5 25
|
anim12i |
⊢ ( ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) → ( 𝑥 : 𝐴 ⟶ 𝐴 ∧ 𝑦 : 𝐴 ⟶ 𝐴 ) ) |
27 |
1 6
|
elefmndbas |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑦 ∈ ( Base ‘ 𝑀 ) ↔ 𝑦 : 𝐴 ⟶ 𝐴 ) ) |
28 |
7 27
|
anbi12d |
⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝑥 ∈ ( Base ‘ 𝑀 ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) ↔ ( 𝑥 : 𝐴 ⟶ 𝐴 ∧ 𝑦 : 𝐴 ⟶ 𝐴 ) ) ) |
29 |
26 28
|
syl5ibr |
⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) → ( 𝑥 ∈ ( Base ‘ 𝑀 ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) ) ) |
30 |
29
|
imp |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) → ( 𝑥 ∈ ( Base ‘ 𝑀 ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) ) |
31 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
32 |
1 6 31
|
efmndov |
⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑀 ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝑥 ∘ 𝑦 ) ) |
33 |
30 32
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝑥 ∘ 𝑦 ) ) |
34 |
33
|
eleq1d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ↔ ( 𝑥 ∘ 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) ) |
35 |
2 19
|
coex |
⊢ ( 𝑥 ∘ 𝑦 ) ∈ V |
36 |
|
foeq1 |
⊢ ( ℎ = ( 𝑥 ∘ 𝑦 ) → ( ℎ : 𝐴 –onto→ 𝐴 ↔ ( 𝑥 ∘ 𝑦 ) : 𝐴 –onto→ 𝐴 ) ) |
37 |
35 36
|
elab |
⊢ ( ( 𝑥 ∘ 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ↔ ( 𝑥 ∘ 𝑦 ) : 𝐴 –onto→ 𝐴 ) |
38 |
34 37
|
bitrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ↔ ( 𝑥 ∘ 𝑦 ) : 𝐴 –onto→ 𝐴 ) ) |
39 |
24 38
|
mpbird |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) |
40 |
39
|
ex |
⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝑥 : 𝐴 –onto→ 𝐴 ∧ 𝑦 : 𝐴 –onto→ 𝐴 ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) ) |
41 |
22 40
|
syl5bi |
⊢ ( 𝐴 ∈ 𝑉 → ( ( 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∧ 𝑦 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) ) |
42 |
41
|
ralrimivv |
⊢ ( 𝐴 ∈ 𝑉 → ∀ 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∀ 𝑦 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) |
43 |
1
|
efmndmnd |
⊢ ( 𝐴 ∈ 𝑉 → 𝑀 ∈ Mnd ) |
44 |
|
eqid |
⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 ) |
45 |
6 44 31
|
issubm |
⊢ ( 𝑀 ∈ Mnd → ( { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∈ ( SubMnd ‘ 𝑀 ) ↔ ( { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∧ ∀ 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∀ 𝑦 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) ) ) |
46 |
43 45
|
syl |
⊢ ( 𝐴 ∈ 𝑉 → ( { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∈ ( SubMnd ‘ 𝑀 ) ↔ ( { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∧ ∀ 𝑥 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∀ 𝑦 ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ∈ { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ) ) ) |
47 |
10 18 42 46
|
mpbir3and |
⊢ ( 𝐴 ∈ 𝑉 → { ℎ ∣ ℎ : 𝐴 –onto→ 𝐴 } ∈ ( SubMnd ‘ 𝑀 ) ) |