Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
|- (/) e. _V |
2 |
|
isust |
|- ( (/) e. _V -> ( u e. ( UnifOn ` (/) ) <-> ( u C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. u /\ A. v e. u ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` (/) ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) ) ) |
3 |
1 2
|
ax-mp |
|- ( u e. ( UnifOn ` (/) ) <-> ( u C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. u /\ A. v e. u ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. u ) /\ A. w e. u ( v i^i w ) e. u /\ ( ( _I |` (/) ) C_ v /\ `' v e. u /\ E. w e. u ( w o. w ) C_ v ) ) ) ) |
4 |
3
|
simp1bi |
|- ( u e. ( UnifOn ` (/) ) -> u C_ ~P ( (/) X. (/) ) ) |
5 |
|
0xp |
|- ( (/) X. (/) ) = (/) |
6 |
5
|
pweqi |
|- ~P ( (/) X. (/) ) = ~P (/) |
7 |
|
pw0 |
|- ~P (/) = { (/) } |
8 |
6 7
|
eqtri |
|- ~P ( (/) X. (/) ) = { (/) } |
9 |
4 8
|
sseqtrdi |
|- ( u e. ( UnifOn ` (/) ) -> u C_ { (/) } ) |
10 |
|
ustbasel |
|- ( u e. ( UnifOn ` (/) ) -> ( (/) X. (/) ) e. u ) |
11 |
5 10
|
eqeltrrid |
|- ( u e. ( UnifOn ` (/) ) -> (/) e. u ) |
12 |
11
|
snssd |
|- ( u e. ( UnifOn ` (/) ) -> { (/) } C_ u ) |
13 |
9 12
|
eqssd |
|- ( u e. ( UnifOn ` (/) ) -> u = { (/) } ) |
14 |
|
velsn |
|- ( u e. { { (/) } } <-> u = { (/) } ) |
15 |
13 14
|
sylibr |
|- ( u e. ( UnifOn ` (/) ) -> u e. { { (/) } } ) |
16 |
15
|
ssriv |
|- ( UnifOn ` (/) ) C_ { { (/) } } |
17 |
8
|
eqimss2i |
|- { (/) } C_ ~P ( (/) X. (/) ) |
18 |
1
|
snid |
|- (/) e. { (/) } |
19 |
5 18
|
eqeltri |
|- ( (/) X. (/) ) e. { (/) } |
20 |
18
|
a1i |
|- ( (/) C_ (/) -> (/) e. { (/) } ) |
21 |
8
|
raleqi |
|- ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) <-> A. w e. { (/) } ( (/) C_ w -> w e. { (/) } ) ) |
22 |
|
sseq2 |
|- ( w = (/) -> ( (/) C_ w <-> (/) C_ (/) ) ) |
23 |
|
eleq1 |
|- ( w = (/) -> ( w e. { (/) } <-> (/) e. { (/) } ) ) |
24 |
22 23
|
imbi12d |
|- ( w = (/) -> ( ( (/) C_ w -> w e. { (/) } ) <-> ( (/) C_ (/) -> (/) e. { (/) } ) ) ) |
25 |
1 24
|
ralsn |
|- ( A. w e. { (/) } ( (/) C_ w -> w e. { (/) } ) <-> ( (/) C_ (/) -> (/) e. { (/) } ) ) |
26 |
21 25
|
bitri |
|- ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) <-> ( (/) C_ (/) -> (/) e. { (/) } ) ) |
27 |
20 26
|
mpbir |
|- A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) |
28 |
|
inidm |
|- ( (/) i^i (/) ) = (/) |
29 |
28 18
|
eqeltri |
|- ( (/) i^i (/) ) e. { (/) } |
30 |
|
ineq2 |
|- ( w = (/) -> ( (/) i^i w ) = ( (/) i^i (/) ) ) |
31 |
30
|
eleq1d |
|- ( w = (/) -> ( ( (/) i^i w ) e. { (/) } <-> ( (/) i^i (/) ) e. { (/) } ) ) |
32 |
1 31
|
ralsn |
|- ( A. w e. { (/) } ( (/) i^i w ) e. { (/) } <-> ( (/) i^i (/) ) e. { (/) } ) |
33 |
29 32
|
mpbir |
|- A. w e. { (/) } ( (/) i^i w ) e. { (/) } |
34 |
|
res0 |
|- ( _I |` (/) ) = (/) |
35 |
34
|
eqimssi |
|- ( _I |` (/) ) C_ (/) |
36 |
|
cnv0 |
|- `' (/) = (/) |
37 |
36 18
|
eqeltri |
|- `' (/) e. { (/) } |
38 |
|
0trrel |
|- ( (/) o. (/) ) C_ (/) |
39 |
|
id |
|- ( w = (/) -> w = (/) ) |
40 |
39 39
|
coeq12d |
|- ( w = (/) -> ( w o. w ) = ( (/) o. (/) ) ) |
41 |
40
|
sseq1d |
|- ( w = (/) -> ( ( w o. w ) C_ (/) <-> ( (/) o. (/) ) C_ (/) ) ) |
42 |
1 41
|
rexsn |
|- ( E. w e. { (/) } ( w o. w ) C_ (/) <-> ( (/) o. (/) ) C_ (/) ) |
43 |
38 42
|
mpbir |
|- E. w e. { (/) } ( w o. w ) C_ (/) |
44 |
35 37 43
|
3pm3.2i |
|- ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) |
45 |
|
sseq1 |
|- ( v = (/) -> ( v C_ w <-> (/) C_ w ) ) |
46 |
45
|
imbi1d |
|- ( v = (/) -> ( ( v C_ w -> w e. { (/) } ) <-> ( (/) C_ w -> w e. { (/) } ) ) ) |
47 |
46
|
ralbidv |
|- ( v = (/) -> ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) <-> A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) ) ) |
48 |
|
ineq1 |
|- ( v = (/) -> ( v i^i w ) = ( (/) i^i w ) ) |
49 |
48
|
eleq1d |
|- ( v = (/) -> ( ( v i^i w ) e. { (/) } <-> ( (/) i^i w ) e. { (/) } ) ) |
50 |
49
|
ralbidv |
|- ( v = (/) -> ( A. w e. { (/) } ( v i^i w ) e. { (/) } <-> A. w e. { (/) } ( (/) i^i w ) e. { (/) } ) ) |
51 |
|
sseq2 |
|- ( v = (/) -> ( ( _I |` (/) ) C_ v <-> ( _I |` (/) ) C_ (/) ) ) |
52 |
|
cnveq |
|- ( v = (/) -> `' v = `' (/) ) |
53 |
52
|
eleq1d |
|- ( v = (/) -> ( `' v e. { (/) } <-> `' (/) e. { (/) } ) ) |
54 |
|
sseq2 |
|- ( v = (/) -> ( ( w o. w ) C_ v <-> ( w o. w ) C_ (/) ) ) |
55 |
54
|
rexbidv |
|- ( v = (/) -> ( E. w e. { (/) } ( w o. w ) C_ v <-> E. w e. { (/) } ( w o. w ) C_ (/) ) ) |
56 |
51 53 55
|
3anbi123d |
|- ( v = (/) -> ( ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) <-> ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) ) ) |
57 |
47 50 56
|
3anbi123d |
|- ( v = (/) -> ( ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) <-> ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( (/) i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) ) ) ) |
58 |
1 57
|
ralsn |
|- ( A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) <-> ( A. w e. ~P ( (/) X. (/) ) ( (/) C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( (/) i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ (/) /\ `' (/) e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ (/) ) ) ) |
59 |
27 33 44 58
|
mpbir3an |
|- A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) |
60 |
|
isust |
|- ( (/) e. _V -> ( { (/) } e. ( UnifOn ` (/) ) <-> ( { (/) } C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. { (/) } /\ A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) ) ) ) |
61 |
1 60
|
ax-mp |
|- ( { (/) } e. ( UnifOn ` (/) ) <-> ( { (/) } C_ ~P ( (/) X. (/) ) /\ ( (/) X. (/) ) e. { (/) } /\ A. v e. { (/) } ( A. w e. ~P ( (/) X. (/) ) ( v C_ w -> w e. { (/) } ) /\ A. w e. { (/) } ( v i^i w ) e. { (/) } /\ ( ( _I |` (/) ) C_ v /\ `' v e. { (/) } /\ E. w e. { (/) } ( w o. w ) C_ v ) ) ) ) |
62 |
17 19 59 61
|
mpbir3an |
|- { (/) } e. ( UnifOn ` (/) ) |
63 |
|
snssi |
|- ( { (/) } e. ( UnifOn ` (/) ) -> { { (/) } } C_ ( UnifOn ` (/) ) ) |
64 |
62 63
|
ax-mp |
|- { { (/) } } C_ ( UnifOn ` (/) ) |
65 |
16 64
|
eqssi |
|- ( UnifOn ` (/) ) = { { (/) } } |