Step |
Hyp |
Ref |
Expression |
1 |
|
f1resfz0f1d.1 |
|- ( ph -> K e. NN0 ) |
2 |
|
f1resfz0f1d.2 |
|- ( ph -> F : ( 0 ... K ) --> V ) |
3 |
|
f1resfz0f1d.3 |
|- ( ph -> ( F |` ( 1 ... K ) ) : ( 1 ... K ) -1-1-> V ) |
4 |
|
f1resfz0f1d.4 |
|- ( ph -> ( ( F " { 0 } ) i^i ( F " ( 1 ... K ) ) ) = (/) ) |
5 |
|
fz1ssfz0 |
|- ( 1 ... K ) C_ ( 0 ... K ) |
6 |
5
|
a1i |
|- ( ph -> ( 1 ... K ) C_ ( 0 ... K ) ) |
7 |
|
0elfz |
|- ( K e. NN0 -> 0 e. ( 0 ... K ) ) |
8 |
|
snssi |
|- ( 0 e. ( 0 ... K ) -> { 0 } C_ ( 0 ... K ) ) |
9 |
1 7 8
|
3syl |
|- ( ph -> { 0 } C_ ( 0 ... K ) ) |
10 |
2 9
|
fssresd |
|- ( ph -> ( F |` { 0 } ) : { 0 } --> V ) |
11 |
|
eqidd |
|- ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` 0 ) -> 0 = 0 ) |
12 |
|
0nn0 |
|- 0 e. NN0 |
13 |
|
fveqeq2 |
|- ( x = 0 -> ( ( ( F |` { 0 } ) ` x ) = ( ( F |` { 0 } ) ` y ) <-> ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` y ) ) ) |
14 |
|
eqeq1 |
|- ( x = 0 -> ( x = y <-> 0 = y ) ) |
15 |
13 14
|
imbi12d |
|- ( x = 0 -> ( ( ( ( F |` { 0 } ) ` x ) = ( ( F |` { 0 } ) ` y ) -> x = y ) <-> ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` y ) -> 0 = y ) ) ) |
16 |
|
fveq2 |
|- ( y = 0 -> ( ( F |` { 0 } ) ` y ) = ( ( F |` { 0 } ) ` 0 ) ) |
17 |
16
|
eqeq2d |
|- ( y = 0 -> ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` y ) <-> ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` 0 ) ) ) |
18 |
|
eqeq2 |
|- ( y = 0 -> ( 0 = y <-> 0 = 0 ) ) |
19 |
17 18
|
imbi12d |
|- ( y = 0 -> ( ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` y ) -> 0 = y ) <-> ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` 0 ) -> 0 = 0 ) ) ) |
20 |
15 19
|
2ralsng |
|- ( ( 0 e. NN0 /\ 0 e. NN0 ) -> ( A. x e. { 0 } A. y e. { 0 } ( ( ( F |` { 0 } ) ` x ) = ( ( F |` { 0 } ) ` y ) -> x = y ) <-> ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` 0 ) -> 0 = 0 ) ) ) |
21 |
12 12 20
|
mp2an |
|- ( A. x e. { 0 } A. y e. { 0 } ( ( ( F |` { 0 } ) ` x ) = ( ( F |` { 0 } ) ` y ) -> x = y ) <-> ( ( ( F |` { 0 } ) ` 0 ) = ( ( F |` { 0 } ) ` 0 ) -> 0 = 0 ) ) |
22 |
11 21
|
mpbir |
|- A. x e. { 0 } A. y e. { 0 } ( ( ( F |` { 0 } ) ` x ) = ( ( F |` { 0 } ) ` y ) -> x = y ) |
23 |
|
dff13 |
|- ( ( F |` { 0 } ) : { 0 } -1-1-> V <-> ( ( F |` { 0 } ) : { 0 } --> V /\ A. x e. { 0 } A. y e. { 0 } ( ( ( F |` { 0 } ) ` x ) = ( ( F |` { 0 } ) ` y ) -> x = y ) ) ) |
24 |
10 22 23
|
sylanblrc |
|- ( ph -> ( F |` { 0 } ) : { 0 } -1-1-> V ) |
25 |
|
uncom |
|- ( ( 1 ... K ) u. { 0 } ) = ( { 0 } u. ( 1 ... K ) ) |
26 |
|
fz0sn0fz1 |
|- ( K e. NN0 -> ( 0 ... K ) = ( { 0 } u. ( 1 ... K ) ) ) |
27 |
1 26
|
syl |
|- ( ph -> ( 0 ... K ) = ( { 0 } u. ( 1 ... K ) ) ) |
28 |
25 27
|
eqtr4id |
|- ( ph -> ( ( 1 ... K ) u. { 0 } ) = ( 0 ... K ) ) |
29 |
|
0nelfz1 |
|- 0 e/ ( 1 ... K ) |
30 |
29
|
neli |
|- -. 0 e. ( 1 ... K ) |
31 |
|
disjsn |
|- ( ( ( 1 ... K ) i^i { 0 } ) = (/) <-> -. 0 e. ( 1 ... K ) ) |
32 |
30 31
|
mpbir |
|- ( ( 1 ... K ) i^i { 0 } ) = (/) |
33 |
|
uneqdifeq |
|- ( ( ( 1 ... K ) C_ ( 0 ... K ) /\ ( ( 1 ... K ) i^i { 0 } ) = (/) ) -> ( ( ( 1 ... K ) u. { 0 } ) = ( 0 ... K ) <-> ( ( 0 ... K ) \ ( 1 ... K ) ) = { 0 } ) ) |
34 |
5 32 33
|
mp2an |
|- ( ( ( 1 ... K ) u. { 0 } ) = ( 0 ... K ) <-> ( ( 0 ... K ) \ ( 1 ... K ) ) = { 0 } ) |
35 |
28 34
|
sylib |
|- ( ph -> ( ( 0 ... K ) \ ( 1 ... K ) ) = { 0 } ) |
36 |
35
|
eqcomd |
|- ( ph -> { 0 } = ( ( 0 ... K ) \ ( 1 ... K ) ) ) |
37 |
36
|
reseq2d |
|- ( ph -> ( F |` { 0 } ) = ( F |` ( ( 0 ... K ) \ ( 1 ... K ) ) ) ) |
38 |
|
eqidd |
|- ( ph -> V = V ) |
39 |
37 36 38
|
f1eq123d |
|- ( ph -> ( ( F |` { 0 } ) : { 0 } -1-1-> V <-> ( F |` ( ( 0 ... K ) \ ( 1 ... K ) ) ) : ( ( 0 ... K ) \ ( 1 ... K ) ) -1-1-> V ) ) |
40 |
24 39
|
mpbid |
|- ( ph -> ( F |` ( ( 0 ... K ) \ ( 1 ... K ) ) ) : ( ( 0 ... K ) \ ( 1 ... K ) ) -1-1-> V ) |
41 |
36
|
imaeq2d |
|- ( ph -> ( F " { 0 } ) = ( F " ( ( 0 ... K ) \ ( 1 ... K ) ) ) ) |
42 |
41
|
ineq2d |
|- ( ph -> ( ( F " ( 1 ... K ) ) i^i ( F " { 0 } ) ) = ( ( F " ( 1 ... K ) ) i^i ( F " ( ( 0 ... K ) \ ( 1 ... K ) ) ) ) ) |
43 |
|
incom |
|- ( ( F " { 0 } ) i^i ( F " ( 1 ... K ) ) ) = ( ( F " ( 1 ... K ) ) i^i ( F " { 0 } ) ) |
44 |
43 4
|
eqtr3id |
|- ( ph -> ( ( F " ( 1 ... K ) ) i^i ( F " { 0 } ) ) = (/) ) |
45 |
42 44
|
eqtr3d |
|- ( ph -> ( ( F " ( 1 ... K ) ) i^i ( F " ( ( 0 ... K ) \ ( 1 ... K ) ) ) ) = (/) ) |
46 |
6 2 3 40 45
|
f1resrcmplf1d |
|- ( ph -> F : ( 0 ... K ) -1-1-> V ) |