Step |
Hyp |
Ref |
Expression |
1 |
|
reltpos |
|- Rel tpos tpos F |
2 |
|
relinxp |
|- Rel ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) |
3 |
|
relcnv |
|- Rel `' dom tpos F |
4 |
|
df-rel |
|- ( Rel `' dom tpos F <-> `' dom tpos F C_ ( _V X. _V ) ) |
5 |
3 4
|
mpbi |
|- `' dom tpos F C_ ( _V X. _V ) |
6 |
|
simpl |
|- ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) -> w e. `' dom tpos F ) |
7 |
5 6
|
sselid |
|- ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) -> w e. ( _V X. _V ) ) |
8 |
|
simpr |
|- ( ( w F z /\ w e. ( _V X. _V ) ) -> w e. ( _V X. _V ) ) |
9 |
|
elvv |
|- ( w e. ( _V X. _V ) <-> E. x E. y w = <. x , y >. ) |
10 |
|
eleq1 |
|- ( w = <. x , y >. -> ( w e. `' dom tpos F <-> <. x , y >. e. `' dom tpos F ) ) |
11 |
|
vex |
|- x e. _V |
12 |
|
vex |
|- y e. _V |
13 |
11 12
|
opelcnv |
|- ( <. x , y >. e. `' dom tpos F <-> <. y , x >. e. dom tpos F ) |
14 |
10 13
|
bitrdi |
|- ( w = <. x , y >. -> ( w e. `' dom tpos F <-> <. y , x >. e. dom tpos F ) ) |
15 |
|
sneq |
|- ( w = <. x , y >. -> { w } = { <. x , y >. } ) |
16 |
15
|
cnveqd |
|- ( w = <. x , y >. -> `' { w } = `' { <. x , y >. } ) |
17 |
16
|
unieqd |
|- ( w = <. x , y >. -> U. `' { w } = U. `' { <. x , y >. } ) |
18 |
|
opswap |
|- U. `' { <. x , y >. } = <. y , x >. |
19 |
17 18
|
eqtrdi |
|- ( w = <. x , y >. -> U. `' { w } = <. y , x >. ) |
20 |
19
|
breq1d |
|- ( w = <. x , y >. -> ( U. `' { w } tpos F z <-> <. y , x >. tpos F z ) ) |
21 |
14 20
|
anbi12d |
|- ( w = <. x , y >. -> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> ( <. y , x >. e. dom tpos F /\ <. y , x >. tpos F z ) ) ) |
22 |
|
opex |
|- <. y , x >. e. _V |
23 |
|
vex |
|- z e. _V |
24 |
22 23
|
breldm |
|- ( <. y , x >. tpos F z -> <. y , x >. e. dom tpos F ) |
25 |
24
|
pm4.71ri |
|- ( <. y , x >. tpos F z <-> ( <. y , x >. e. dom tpos F /\ <. y , x >. tpos F z ) ) |
26 |
|
brtpos |
|- ( z e. _V -> ( <. y , x >. tpos F z <-> <. x , y >. F z ) ) |
27 |
26
|
elv |
|- ( <. y , x >. tpos F z <-> <. x , y >. F z ) |
28 |
25 27
|
bitr3i |
|- ( ( <. y , x >. e. dom tpos F /\ <. y , x >. tpos F z ) <-> <. x , y >. F z ) |
29 |
21 28
|
bitrdi |
|- ( w = <. x , y >. -> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> <. x , y >. F z ) ) |
30 |
|
breq1 |
|- ( w = <. x , y >. -> ( w F z <-> <. x , y >. F z ) ) |
31 |
29 30
|
bitr4d |
|- ( w = <. x , y >. -> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> w F z ) ) |
32 |
31
|
exlimivv |
|- ( E. x E. y w = <. x , y >. -> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> w F z ) ) |
33 |
9 32
|
sylbi |
|- ( w e. ( _V X. _V ) -> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> w F z ) ) |
34 |
|
iba |
|- ( w e. ( _V X. _V ) -> ( w F z <-> ( w F z /\ w e. ( _V X. _V ) ) ) ) |
35 |
33 34
|
bitrd |
|- ( w e. ( _V X. _V ) -> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> ( w F z /\ w e. ( _V X. _V ) ) ) ) |
36 |
7 8 35
|
pm5.21nii |
|- ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) <-> ( w F z /\ w e. ( _V X. _V ) ) ) |
37 |
|
elsni |
|- ( w e. { (/) } -> w = (/) ) |
38 |
37
|
sneqd |
|- ( w e. { (/) } -> { w } = { (/) } ) |
39 |
38
|
cnveqd |
|- ( w e. { (/) } -> `' { w } = `' { (/) } ) |
40 |
|
cnvsn0 |
|- `' { (/) } = (/) |
41 |
39 40
|
eqtrdi |
|- ( w e. { (/) } -> `' { w } = (/) ) |
42 |
41
|
unieqd |
|- ( w e. { (/) } -> U. `' { w } = U. (/) ) |
43 |
|
uni0 |
|- U. (/) = (/) |
44 |
42 43
|
eqtrdi |
|- ( w e. { (/) } -> U. `' { w } = (/) ) |
45 |
44
|
breq1d |
|- ( w e. { (/) } -> ( U. `' { w } tpos F z <-> (/) tpos F z ) ) |
46 |
|
brtpos0 |
|- ( z e. _V -> ( (/) tpos F z <-> (/) F z ) ) |
47 |
46
|
elv |
|- ( (/) tpos F z <-> (/) F z ) |
48 |
45 47
|
bitrdi |
|- ( w e. { (/) } -> ( U. `' { w } tpos F z <-> (/) F z ) ) |
49 |
37
|
breq1d |
|- ( w e. { (/) } -> ( w F z <-> (/) F z ) ) |
50 |
48 49
|
bitr4d |
|- ( w e. { (/) } -> ( U. `' { w } tpos F z <-> w F z ) ) |
51 |
50
|
pm5.32i |
|- ( ( w e. { (/) } /\ U. `' { w } tpos F z ) <-> ( w e. { (/) } /\ w F z ) ) |
52 |
51
|
biancomi |
|- ( ( w e. { (/) } /\ U. `' { w } tpos F z ) <-> ( w F z /\ w e. { (/) } ) ) |
53 |
36 52
|
orbi12i |
|- ( ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) \/ ( w e. { (/) } /\ U. `' { w } tpos F z ) ) <-> ( ( w F z /\ w e. ( _V X. _V ) ) \/ ( w F z /\ w e. { (/) } ) ) ) |
54 |
|
andir |
|- ( ( ( w e. `' dom tpos F \/ w e. { (/) } ) /\ U. `' { w } tpos F z ) <-> ( ( w e. `' dom tpos F /\ U. `' { w } tpos F z ) \/ ( w e. { (/) } /\ U. `' { w } tpos F z ) ) ) |
55 |
|
andi |
|- ( ( w F z /\ ( w e. ( _V X. _V ) \/ w e. { (/) } ) ) <-> ( ( w F z /\ w e. ( _V X. _V ) ) \/ ( w F z /\ w e. { (/) } ) ) ) |
56 |
53 54 55
|
3bitr4i |
|- ( ( ( w e. `' dom tpos F \/ w e. { (/) } ) /\ U. `' { w } tpos F z ) <-> ( w F z /\ ( w e. ( _V X. _V ) \/ w e. { (/) } ) ) ) |
57 |
|
elun |
|- ( w e. ( `' dom tpos F u. { (/) } ) <-> ( w e. `' dom tpos F \/ w e. { (/) } ) ) |
58 |
57
|
anbi1i |
|- ( ( w e. ( `' dom tpos F u. { (/) } ) /\ U. `' { w } tpos F z ) <-> ( ( w e. `' dom tpos F \/ w e. { (/) } ) /\ U. `' { w } tpos F z ) ) |
59 |
|
brxp |
|- ( w ( ( ( _V X. _V ) u. { (/) } ) X. _V ) z <-> ( w e. ( ( _V X. _V ) u. { (/) } ) /\ z e. _V ) ) |
60 |
23 59
|
mpbiran2 |
|- ( w ( ( ( _V X. _V ) u. { (/) } ) X. _V ) z <-> w e. ( ( _V X. _V ) u. { (/) } ) ) |
61 |
|
elun |
|- ( w e. ( ( _V X. _V ) u. { (/) } ) <-> ( w e. ( _V X. _V ) \/ w e. { (/) } ) ) |
62 |
60 61
|
bitri |
|- ( w ( ( ( _V X. _V ) u. { (/) } ) X. _V ) z <-> ( w e. ( _V X. _V ) \/ w e. { (/) } ) ) |
63 |
62
|
anbi2i |
|- ( ( w F z /\ w ( ( ( _V X. _V ) u. { (/) } ) X. _V ) z ) <-> ( w F z /\ ( w e. ( _V X. _V ) \/ w e. { (/) } ) ) ) |
64 |
56 58 63
|
3bitr4i |
|- ( ( w e. ( `' dom tpos F u. { (/) } ) /\ U. `' { w } tpos F z ) <-> ( w F z /\ w ( ( ( _V X. _V ) u. { (/) } ) X. _V ) z ) ) |
65 |
|
brtpos2 |
|- ( z e. _V -> ( w tpos tpos F z <-> ( w e. ( `' dom tpos F u. { (/) } ) /\ U. `' { w } tpos F z ) ) ) |
66 |
65
|
elv |
|- ( w tpos tpos F z <-> ( w e. ( `' dom tpos F u. { (/) } ) /\ U. `' { w } tpos F z ) ) |
67 |
|
brin |
|- ( w ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) z <-> ( w F z /\ w ( ( ( _V X. _V ) u. { (/) } ) X. _V ) z ) ) |
68 |
64 66 67
|
3bitr4i |
|- ( w tpos tpos F z <-> w ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) z ) |
69 |
1 2 68
|
eqbrriv |
|- tpos tpos F = ( F i^i ( ( ( _V X. _V ) u. { (/) } ) X. _V ) ) |