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 C_ y |
14 |
11 4
|
cfv |
|- ( f ` x ) |
15 |
12 4
|
cfv |
|- ( f ` y ) |
16 |
14 15
|
wss |
|- ( f ` x ) C_ ( f ` y ) |
17 |
13 16
|
wb |
|- ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) |
18 |
17 10 7
|
wral |
|- A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) |
19 |
18 9 7
|
wral |
|- A. x e. ( PSubSp ` k ) A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) |
20 |
8 19
|
wa |
|- ( f : ( PSubSp ` k ) -1-1-onto-> ( PSubSp ` k ) /\ A. x e. ( PSubSp ` k ) A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) ) |
21 |
20 3
|
cab |
|- { f | ( f : ( PSubSp ` k ) -1-1-onto-> ( PSubSp ` k ) /\ A. x e. ( PSubSp ` k ) A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) ) } |
22 |
1 2 21
|
cmpt |
|- ( k e. _V |-> { f | ( f : ( PSubSp ` k ) -1-1-onto-> ( PSubSp ` k ) /\ A. x e. ( PSubSp ` k ) A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) ) } ) |
23 |
0 22
|
wceq |
|- PAut = ( k e. _V |-> { f | ( f : ( PSubSp ` k ) -1-1-onto-> ( PSubSp ` k ) /\ A. x e. ( PSubSp ` k ) A. y e. ( PSubSp ` k ) ( x C_ y <-> ( f ` x ) C_ ( f ` y ) ) ) } ) |