Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
|- (/) e. _V |
2 |
1
|
eldm |
|- ( (/) e. dom F <-> E. y (/) F y ) |
3 |
|
brtpos0 |
|- ( y e. _V -> ( (/) tpos F y <-> (/) F y ) ) |
4 |
3
|
elv |
|- ( (/) tpos F y <-> (/) F y ) |
5 |
|
0nelrel0 |
|- ( Rel dom tpos F -> -. (/) e. dom tpos F ) |
6 |
|
vex |
|- y e. _V |
7 |
1 6
|
breldm |
|- ( (/) tpos F y -> (/) e. dom tpos F ) |
8 |
5 7
|
nsyl3 |
|- ( (/) tpos F y -> -. Rel dom tpos F ) |
9 |
4 8
|
sylbir |
|- ( (/) F y -> -. Rel dom tpos F ) |
10 |
9
|
exlimiv |
|- ( E. y (/) F y -> -. Rel dom tpos F ) |
11 |
2 10
|
sylbi |
|- ( (/) e. dom F -> -. Rel dom tpos F ) |
12 |
11
|
con2i |
|- ( Rel dom tpos F -> -. (/) e. dom F ) |
13 |
|
vex |
|- x e. _V |
14 |
13
|
eldm |
|- ( x e. dom tpos F <-> E. y x tpos F y ) |
15 |
|
relcnv |
|- Rel `' dom F |
16 |
|
df-rel |
|- ( Rel `' dom F <-> `' dom F C_ ( _V X. _V ) ) |
17 |
15 16
|
mpbi |
|- `' dom F C_ ( _V X. _V ) |
18 |
17
|
sseli |
|- ( x e. `' dom F -> x e. ( _V X. _V ) ) |
19 |
18
|
a1i |
|- ( ( -. (/) e. dom F /\ x tpos F y ) -> ( x e. `' dom F -> x e. ( _V X. _V ) ) ) |
20 |
|
elsni |
|- ( x e. { (/) } -> x = (/) ) |
21 |
20
|
breq1d |
|- ( x e. { (/) } -> ( x tpos F y <-> (/) tpos F y ) ) |
22 |
1 6
|
breldm |
|- ( (/) F y -> (/) e. dom F ) |
23 |
22
|
pm2.24d |
|- ( (/) F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) ) |
24 |
4 23
|
sylbi |
|- ( (/) tpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) ) |
25 |
21 24
|
syl6bi |
|- ( x e. { (/) } -> ( x tpos F y -> ( -. (/) e. dom F -> x e. ( _V X. _V ) ) ) ) |
26 |
25
|
com3l |
|- ( x tpos F y -> ( -. (/) e. dom F -> ( x e. { (/) } -> x e. ( _V X. _V ) ) ) ) |
27 |
26
|
impcom |
|- ( ( -. (/) e. dom F /\ x tpos F y ) -> ( x e. { (/) } -> x e. ( _V X. _V ) ) ) |
28 |
|
brtpos2 |
|- ( y e. _V -> ( x tpos F y <-> ( x e. ( `' dom F u. { (/) } ) /\ U. `' { x } F y ) ) ) |
29 |
6 28
|
ax-mp |
|- ( x tpos F y <-> ( x e. ( `' dom F u. { (/) } ) /\ U. `' { x } F y ) ) |
30 |
29
|
simplbi |
|- ( x tpos F y -> x e. ( `' dom F u. { (/) } ) ) |
31 |
|
elun |
|- ( x e. ( `' dom F u. { (/) } ) <-> ( x e. `' dom F \/ x e. { (/) } ) ) |
32 |
30 31
|
sylib |
|- ( x tpos F y -> ( x e. `' dom F \/ x e. { (/) } ) ) |
33 |
32
|
adantl |
|- ( ( -. (/) e. dom F /\ x tpos F y ) -> ( x e. `' dom F \/ x e. { (/) } ) ) |
34 |
19 27 33
|
mpjaod |
|- ( ( -. (/) e. dom F /\ x tpos F y ) -> x e. ( _V X. _V ) ) |
35 |
34
|
ex |
|- ( -. (/) e. dom F -> ( x tpos F y -> x e. ( _V X. _V ) ) ) |
36 |
35
|
exlimdv |
|- ( -. (/) e. dom F -> ( E. y x tpos F y -> x e. ( _V X. _V ) ) ) |
37 |
14 36
|
syl5bi |
|- ( -. (/) e. dom F -> ( x e. dom tpos F -> x e. ( _V X. _V ) ) ) |
38 |
37
|
ssrdv |
|- ( -. (/) e. dom F -> dom tpos F C_ ( _V X. _V ) ) |
39 |
|
df-rel |
|- ( Rel dom tpos F <-> dom tpos F C_ ( _V X. _V ) ) |
40 |
38 39
|
sylibr |
|- ( -. (/) e. dom F -> Rel dom tpos F ) |
41 |
12 40
|
impbii |
|- ( Rel dom tpos F <-> -. (/) e. dom F ) |