Step |
Hyp |
Ref |
Expression |
0 |
|
cpautN |
⊢ PAut |
1 |
|
vk |
⊢ 𝑘 |
2 |
|
cvv |
⊢ V |
3 |
|
vf |
⊢ 𝑓 |
4 |
3
|
cv |
⊢ 𝑓 |
5 |
|
cpsubsp |
⊢ PSubSp |
6 |
1
|
cv |
⊢ 𝑘 |
7 |
6 5
|
cfv |
⊢ ( PSubSp ‘ 𝑘 ) |
8 |
7 7 4
|
wf1o |
⊢ 𝑓 : ( PSubSp ‘ 𝑘 ) –1-1-onto→ ( PSubSp ‘ 𝑘 ) |
9 |
|
vx |
⊢ 𝑥 |
10 |
|
vy |
⊢ 𝑦 |
11 |
9
|
cv |
⊢ 𝑥 |
12 |
10
|
cv |
⊢ 𝑦 |
13 |
11 12
|
wss |
⊢ 𝑥 ⊆ 𝑦 |
14 |
11 4
|
cfv |
⊢ ( 𝑓 ‘ 𝑥 ) |
15 |
12 4
|
cfv |
⊢ ( 𝑓 ‘ 𝑦 ) |
16 |
14 15
|
wss |
⊢ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) |
17 |
13 16
|
wb |
⊢ ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) |
18 |
17 10 7
|
wral |
⊢ ∀ 𝑦 ∈ ( PSubSp ‘ 𝑘 ) ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) |
19 |
18 9 7
|
wral |
⊢ ∀ 𝑥 ∈ ( PSubSp ‘ 𝑘 ) ∀ 𝑦 ∈ ( PSubSp ‘ 𝑘 ) ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) |
20 |
8 19
|
wa |
⊢ ( 𝑓 : ( PSubSp ‘ 𝑘 ) –1-1-onto→ ( PSubSp ‘ 𝑘 ) ∧ ∀ 𝑥 ∈ ( PSubSp ‘ 𝑘 ) ∀ 𝑦 ∈ ( PSubSp ‘ 𝑘 ) ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) |
21 |
20 3
|
cab |
⊢ { 𝑓 ∣ ( 𝑓 : ( PSubSp ‘ 𝑘 ) –1-1-onto→ ( PSubSp ‘ 𝑘 ) ∧ ∀ 𝑥 ∈ ( PSubSp ‘ 𝑘 ) ∀ 𝑦 ∈ ( PSubSp ‘ 𝑘 ) ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) } |
22 |
1 2 21
|
cmpt |
⊢ ( 𝑘 ∈ V ↦ { 𝑓 ∣ ( 𝑓 : ( PSubSp ‘ 𝑘 ) –1-1-onto→ ( PSubSp ‘ 𝑘 ) ∧ ∀ 𝑥 ∈ ( PSubSp ‘ 𝑘 ) ∀ 𝑦 ∈ ( PSubSp ‘ 𝑘 ) ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) } ) |
23 |
0 22
|
wceq |
⊢ PAut = ( 𝑘 ∈ V ↦ { 𝑓 ∣ ( 𝑓 : ( PSubSp ‘ 𝑘 ) –1-1-onto→ ( PSubSp ‘ 𝑘 ) ∧ ∀ 𝑥 ∈ ( PSubSp ‘ 𝑘 ) ∀ 𝑦 ∈ ( PSubSp ‘ 𝑘 ) ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) } ) |