Step |
Hyp |
Ref |
Expression |
1 |
|
clscld.1 |
|- X = U. J |
2 |
|
df-rab |
|- { z e. ( Clsd ` J ) | S C_ z } = { z | ( z e. ( Clsd ` J ) /\ S C_ z ) } |
3 |
1
|
cldopn |
|- ( z e. ( Clsd ` J ) -> ( X \ z ) e. J ) |
4 |
3
|
ad2antrl |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> ( X \ z ) e. J ) |
5 |
|
sscon |
|- ( S C_ z -> ( X \ z ) C_ ( X \ S ) ) |
6 |
5
|
ad2antll |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> ( X \ z ) C_ ( X \ S ) ) |
7 |
1
|
topopn |
|- ( J e. Top -> X e. J ) |
8 |
|
difexg |
|- ( X e. J -> ( X \ z ) e. _V ) |
9 |
|
elpwg |
|- ( ( X \ z ) e. _V -> ( ( X \ z ) e. ~P ( X \ S ) <-> ( X \ z ) C_ ( X \ S ) ) ) |
10 |
7 8 9
|
3syl |
|- ( J e. Top -> ( ( X \ z ) e. ~P ( X \ S ) <-> ( X \ z ) C_ ( X \ S ) ) ) |
11 |
10
|
ad2antrr |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> ( ( X \ z ) e. ~P ( X \ S ) <-> ( X \ z ) C_ ( X \ S ) ) ) |
12 |
6 11
|
mpbird |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> ( X \ z ) e. ~P ( X \ S ) ) |
13 |
4 12
|
elind |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> ( X \ z ) e. ( J i^i ~P ( X \ S ) ) ) |
14 |
1
|
cldss |
|- ( z e. ( Clsd ` J ) -> z C_ X ) |
15 |
14
|
ad2antrl |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> z C_ X ) |
16 |
|
dfss4 |
|- ( z C_ X <-> ( X \ ( X \ z ) ) = z ) |
17 |
15 16
|
sylib |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> ( X \ ( X \ z ) ) = z ) |
18 |
17
|
eqcomd |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> z = ( X \ ( X \ z ) ) ) |
19 |
|
difeq2 |
|- ( x = ( X \ z ) -> ( X \ x ) = ( X \ ( X \ z ) ) ) |
20 |
19
|
rspceeqv |
|- ( ( ( X \ z ) e. ( J i^i ~P ( X \ S ) ) /\ z = ( X \ ( X \ z ) ) ) -> E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) ) |
21 |
13 18 20
|
syl2anc |
|- ( ( ( J e. Top /\ S C_ X ) /\ ( z e. ( Clsd ` J ) /\ S C_ z ) ) -> E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) ) |
22 |
21
|
ex |
|- ( ( J e. Top /\ S C_ X ) -> ( ( z e. ( Clsd ` J ) /\ S C_ z ) -> E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) ) ) |
23 |
|
simpl |
|- ( ( J e. Top /\ S C_ X ) -> J e. Top ) |
24 |
|
elinel1 |
|- ( x e. ( J i^i ~P ( X \ S ) ) -> x e. J ) |
25 |
1
|
opncld |
|- ( ( J e. Top /\ x e. J ) -> ( X \ x ) e. ( Clsd ` J ) ) |
26 |
23 24 25
|
syl2an |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> ( X \ x ) e. ( Clsd ` J ) ) |
27 |
|
elinel2 |
|- ( x e. ( J i^i ~P ( X \ S ) ) -> x e. ~P ( X \ S ) ) |
28 |
27
|
adantl |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> x e. ~P ( X \ S ) ) |
29 |
28
|
elpwid |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> x C_ ( X \ S ) ) |
30 |
29
|
difss2d |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> x C_ X ) |
31 |
|
simplr |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> S C_ X ) |
32 |
|
ssconb |
|- ( ( x C_ X /\ S C_ X ) -> ( x C_ ( X \ S ) <-> S C_ ( X \ x ) ) ) |
33 |
30 31 32
|
syl2anc |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> ( x C_ ( X \ S ) <-> S C_ ( X \ x ) ) ) |
34 |
29 33
|
mpbid |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> S C_ ( X \ x ) ) |
35 |
26 34
|
jca |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> ( ( X \ x ) e. ( Clsd ` J ) /\ S C_ ( X \ x ) ) ) |
36 |
|
eleq1 |
|- ( z = ( X \ x ) -> ( z e. ( Clsd ` J ) <-> ( X \ x ) e. ( Clsd ` J ) ) ) |
37 |
|
sseq2 |
|- ( z = ( X \ x ) -> ( S C_ z <-> S C_ ( X \ x ) ) ) |
38 |
36 37
|
anbi12d |
|- ( z = ( X \ x ) -> ( ( z e. ( Clsd ` J ) /\ S C_ z ) <-> ( ( X \ x ) e. ( Clsd ` J ) /\ S C_ ( X \ x ) ) ) ) |
39 |
35 38
|
syl5ibrcom |
|- ( ( ( J e. Top /\ S C_ X ) /\ x e. ( J i^i ~P ( X \ S ) ) ) -> ( z = ( X \ x ) -> ( z e. ( Clsd ` J ) /\ S C_ z ) ) ) |
40 |
39
|
rexlimdva |
|- ( ( J e. Top /\ S C_ X ) -> ( E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) -> ( z e. ( Clsd ` J ) /\ S C_ z ) ) ) |
41 |
22 40
|
impbid |
|- ( ( J e. Top /\ S C_ X ) -> ( ( z e. ( Clsd ` J ) /\ S C_ z ) <-> E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) ) ) |
42 |
41
|
abbidv |
|- ( ( J e. Top /\ S C_ X ) -> { z | ( z e. ( Clsd ` J ) /\ S C_ z ) } = { z | E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) } ) |
43 |
2 42
|
eqtrid |
|- ( ( J e. Top /\ S C_ X ) -> { z e. ( Clsd ` J ) | S C_ z } = { z | E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) } ) |
44 |
43
|
inteqd |
|- ( ( J e. Top /\ S C_ X ) -> |^| { z e. ( Clsd ` J ) | S C_ z } = |^| { z | E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) } ) |
45 |
|
difexg |
|- ( X e. J -> ( X \ x ) e. _V ) |
46 |
45
|
ralrimivw |
|- ( X e. J -> A. x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) e. _V ) |
47 |
|
dfiin2g |
|- ( A. x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) e. _V -> |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) = |^| { z | E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) } ) |
48 |
7 46 47
|
3syl |
|- ( J e. Top -> |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) = |^| { z | E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) } ) |
49 |
48
|
adantr |
|- ( ( J e. Top /\ S C_ X ) -> |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) = |^| { z | E. x e. ( J i^i ~P ( X \ S ) ) z = ( X \ x ) } ) |
50 |
44 49
|
eqtr4d |
|- ( ( J e. Top /\ S C_ X ) -> |^| { z e. ( Clsd ` J ) | S C_ z } = |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) ) |
51 |
1
|
clsval |
|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { z e. ( Clsd ` J ) | S C_ z } ) |
52 |
|
uniiun |
|- U. ( J i^i ~P ( X \ S ) ) = U_ x e. ( J i^i ~P ( X \ S ) ) x |
53 |
52
|
difeq2i |
|- ( X \ U. ( J i^i ~P ( X \ S ) ) ) = ( X \ U_ x e. ( J i^i ~P ( X \ S ) ) x ) |
54 |
53
|
a1i |
|- ( ( J e. Top /\ S C_ X ) -> ( X \ U. ( J i^i ~P ( X \ S ) ) ) = ( X \ U_ x e. ( J i^i ~P ( X \ S ) ) x ) ) |
55 |
|
0opn |
|- ( J e. Top -> (/) e. J ) |
56 |
55
|
adantr |
|- ( ( J e. Top /\ S C_ X ) -> (/) e. J ) |
57 |
|
0elpw |
|- (/) e. ~P ( X \ S ) |
58 |
57
|
a1i |
|- ( ( J e. Top /\ S C_ X ) -> (/) e. ~P ( X \ S ) ) |
59 |
56 58
|
elind |
|- ( ( J e. Top /\ S C_ X ) -> (/) e. ( J i^i ~P ( X \ S ) ) ) |
60 |
|
ne0i |
|- ( (/) e. ( J i^i ~P ( X \ S ) ) -> ( J i^i ~P ( X \ S ) ) =/= (/) ) |
61 |
|
iindif2 |
|- ( ( J i^i ~P ( X \ S ) ) =/= (/) -> |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) = ( X \ U_ x e. ( J i^i ~P ( X \ S ) ) x ) ) |
62 |
59 60 61
|
3syl |
|- ( ( J e. Top /\ S C_ X ) -> |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) = ( X \ U_ x e. ( J i^i ~P ( X \ S ) ) x ) ) |
63 |
54 62
|
eqtr4d |
|- ( ( J e. Top /\ S C_ X ) -> ( X \ U. ( J i^i ~P ( X \ S ) ) ) = |^|_ x e. ( J i^i ~P ( X \ S ) ) ( X \ x ) ) |
64 |
50 51 63
|
3eqtr4d |
|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( X \ U. ( J i^i ~P ( X \ S ) ) ) ) |
65 |
|
difssd |
|- ( S C_ X -> ( X \ S ) C_ X ) |
66 |
1
|
ntrval |
|- ( ( J e. Top /\ ( X \ S ) C_ X ) -> ( ( int ` J ) ` ( X \ S ) ) = U. ( J i^i ~P ( X \ S ) ) ) |
67 |
65 66
|
sylan2 |
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` ( X \ S ) ) = U. ( J i^i ~P ( X \ S ) ) ) |
68 |
67
|
difeq2d |
|- ( ( J e. Top /\ S C_ X ) -> ( X \ ( ( int ` J ) ` ( X \ S ) ) ) = ( X \ U. ( J i^i ~P ( X \ S ) ) ) ) |
69 |
64 68
|
eqtr4d |
|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( X \ ( ( int ` J ) ` ( X \ S ) ) ) ) |