Step |
Hyp |
Ref |
Expression |
1 |
|
idressubmefmnd.g |
⊢ 𝐺 = ( EndoFMnd ‘ 𝐴 ) |
2 |
|
idresefmnd.e |
⊢ 𝐸 = ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) |
3 |
1
|
idressubmefmnd |
⊢ ( 𝐴 ∈ 𝑉 → { ( I ↾ 𝐴 ) } ∈ ( SubMnd ‘ 𝐺 ) ) |
4 |
1
|
efmndmnd |
⊢ ( 𝐴 ∈ 𝑉 → 𝐺 ∈ Mnd ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
6 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
7 |
|
eqid |
⊢ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) = ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) |
8 |
5 6 7
|
issubm2 |
⊢ ( 𝐺 ∈ Mnd → ( { ( I ↾ 𝐴 ) } ∈ ( SubMnd ‘ 𝐺 ) ↔ ( { ( I ↾ 𝐴 ) } ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ { ( I ↾ 𝐴 ) } ∧ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd ) ) ) |
9 |
4 8
|
syl |
⊢ ( 𝐴 ∈ 𝑉 → ( { ( I ↾ 𝐴 ) } ∈ ( SubMnd ‘ 𝐺 ) ↔ ( { ( I ↾ 𝐴 ) } ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ { ( I ↾ 𝐴 ) } ∧ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd ) ) ) |
10 |
|
snex |
⊢ { ( I ↾ 𝐴 ) } ∈ V |
11 |
2 5
|
ressbas |
⊢ ( { ( I ↾ 𝐴 ) } ∈ V → ( { ( I ↾ 𝐴 ) } ∩ ( Base ‘ 𝐺 ) ) = ( Base ‘ 𝐸 ) ) |
12 |
10 11
|
mp1i |
⊢ ( 𝐴 ∈ 𝑉 → ( { ( I ↾ 𝐴 ) } ∩ ( Base ‘ 𝐺 ) ) = ( Base ‘ 𝐸 ) ) |
13 |
|
inss2 |
⊢ ( { ( I ↾ 𝐴 ) } ∩ ( Base ‘ 𝐺 ) ) ⊆ ( Base ‘ 𝐺 ) |
14 |
12 13
|
eqsstrrdi |
⊢ ( 𝐴 ∈ 𝑉 → ( Base ‘ 𝐸 ) ⊆ ( Base ‘ 𝐺 ) ) |
15 |
2
|
eqcomi |
⊢ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) = 𝐸 |
16 |
15
|
eleq1i |
⊢ ( ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd ↔ 𝐸 ∈ Mnd ) |
17 |
16
|
biimpi |
⊢ ( ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd → 𝐸 ∈ Mnd ) |
18 |
17
|
3ad2ant3 |
⊢ ( ( { ( I ↾ 𝐴 ) } ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ { ( I ↾ 𝐴 ) } ∧ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd ) → 𝐸 ∈ Mnd ) |
19 |
14 18
|
anim12ci |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( { ( I ↾ 𝐴 ) } ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ { ( I ↾ 𝐴 ) } ∧ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd ) ) → ( 𝐸 ∈ Mnd ∧ ( Base ‘ 𝐸 ) ⊆ ( Base ‘ 𝐺 ) ) ) |
20 |
19
|
ex |
⊢ ( 𝐴 ∈ 𝑉 → ( ( { ( I ↾ 𝐴 ) } ⊆ ( Base ‘ 𝐺 ) ∧ ( 0g ‘ 𝐺 ) ∈ { ( I ↾ 𝐴 ) } ∧ ( 𝐺 ↾s { ( I ↾ 𝐴 ) } ) ∈ Mnd ) → ( 𝐸 ∈ Mnd ∧ ( Base ‘ 𝐸 ) ⊆ ( Base ‘ 𝐺 ) ) ) ) |
21 |
9 20
|
sylbid |
⊢ ( 𝐴 ∈ 𝑉 → ( { ( I ↾ 𝐴 ) } ∈ ( SubMnd ‘ 𝐺 ) → ( 𝐸 ∈ Mnd ∧ ( Base ‘ 𝐸 ) ⊆ ( Base ‘ 𝐺 ) ) ) ) |
22 |
3 21
|
mpd |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐸 ∈ Mnd ∧ ( Base ‘ 𝐸 ) ⊆ ( Base ‘ 𝐺 ) ) ) |