Step |
Hyp |
Ref |
Expression |
1 |
|
derang.d |
|- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) ) |
2 |
|
0fin |
|- (/) e. Fin |
3 |
1
|
derangval |
|- ( (/) e. Fin -> ( D ` (/) ) = ( # ` { f | ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) } ) ) |
4 |
2 3
|
ax-mp |
|- ( D ` (/) ) = ( # ` { f | ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) } ) |
5 |
|
ral0 |
|- A. y e. (/) ( f ` y ) =/= y |
6 |
5
|
biantru |
|- ( f : (/) -1-1-onto-> (/) <-> ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) ) |
7 |
|
eqid |
|- (/) = (/) |
8 |
|
f1o00 |
|- ( f : (/) -1-1-onto-> (/) <-> ( f = (/) /\ (/) = (/) ) ) |
9 |
7 8
|
mpbiran2 |
|- ( f : (/) -1-1-onto-> (/) <-> f = (/) ) |
10 |
6 9
|
bitr3i |
|- ( ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) <-> f = (/) ) |
11 |
10
|
abbii |
|- { f | ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) } = { f | f = (/) } |
12 |
|
df-sn |
|- { (/) } = { f | f = (/) } |
13 |
11 12
|
eqtr4i |
|- { f | ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) } = { (/) } |
14 |
13
|
fveq2i |
|- ( # ` { f | ( f : (/) -1-1-onto-> (/) /\ A. y e. (/) ( f ` y ) =/= y ) } ) = ( # ` { (/) } ) |
15 |
|
0ex |
|- (/) e. _V |
16 |
|
hashsng |
|- ( (/) e. _V -> ( # ` { (/) } ) = 1 ) |
17 |
15 16
|
ax-mp |
|- ( # ` { (/) } ) = 1 |
18 |
4 14 17
|
3eqtri |
|- ( D ` (/) ) = 1 |