Step |
Hyp |
Ref |
Expression |
1 |
|
utoptop.1 |
|- J = ( unifTop ` U ) |
2 |
|
relxp |
|- Rel ( X X. X ) |
3 |
|
utoptop |
|- ( U e. ( UnifOn ` X ) -> ( unifTop ` U ) e. Top ) |
4 |
1 3
|
eqeltrid |
|- ( U e. ( UnifOn ` X ) -> J e. Top ) |
5 |
|
txtop |
|- ( ( J e. Top /\ J e. Top ) -> ( J tX J ) e. Top ) |
6 |
4 4 5
|
syl2anc |
|- ( U e. ( UnifOn ` X ) -> ( J tX J ) e. Top ) |
7 |
6
|
ad3antrrr |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( J tX J ) e. Top ) |
8 |
|
simpllr |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> M C_ ( X X. X ) ) |
9 |
|
utoptopon |
|- ( U e. ( UnifOn ` X ) -> ( unifTop ` U ) e. ( TopOn ` X ) ) |
10 |
1 9
|
eqeltrid |
|- ( U e. ( UnifOn ` X ) -> J e. ( TopOn ` X ) ) |
11 |
|
toponuni |
|- ( J e. ( TopOn ` X ) -> X = U. J ) |
12 |
10 11
|
syl |
|- ( U e. ( UnifOn ` X ) -> X = U. J ) |
13 |
12
|
sqxpeqd |
|- ( U e. ( UnifOn ` X ) -> ( X X. X ) = ( U. J X. U. J ) ) |
14 |
|
eqid |
|- U. J = U. J |
15 |
14 14
|
txuni |
|- ( ( J e. Top /\ J e. Top ) -> ( U. J X. U. J ) = U. ( J tX J ) ) |
16 |
4 4 15
|
syl2anc |
|- ( U e. ( UnifOn ` X ) -> ( U. J X. U. J ) = U. ( J tX J ) ) |
17 |
13 16
|
eqtrd |
|- ( U e. ( UnifOn ` X ) -> ( X X. X ) = U. ( J tX J ) ) |
18 |
17
|
ad3antrrr |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( X X. X ) = U. ( J tX J ) ) |
19 |
8 18
|
sseqtrd |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> M C_ U. ( J tX J ) ) |
20 |
|
eqid |
|- U. ( J tX J ) = U. ( J tX J ) |
21 |
20
|
clsss3 |
|- ( ( ( J tX J ) e. Top /\ M C_ U. ( J tX J ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ U. ( J tX J ) ) |
22 |
7 19 21
|
syl2anc |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ U. ( J tX J ) ) |
23 |
22 18
|
sseqtrrd |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ ( X X. X ) ) |
24 |
|
simpr |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z e. ( ( cls ` ( J tX J ) ) ` M ) ) |
25 |
23 24
|
sseldd |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z e. ( X X. X ) ) |
26 |
|
1st2nd |
|- ( ( Rel ( X X. X ) /\ z e. ( X X. X ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
27 |
2 25 26
|
sylancr |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
28 |
|
simp-4l |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> U e. ( UnifOn ` X ) ) |
29 |
|
simpr1l |
|- ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( ( V e. U /\ `' V = V ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) ) -> V e. U ) |
30 |
29
|
3anassrs |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> V e. U ) |
31 |
|
ustrel |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U ) -> Rel V ) |
32 |
28 30 31
|
syl2anc |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> Rel V ) |
33 |
|
simpr |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) |
34 |
|
elin |
|- ( r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) <-> ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) /\ r e. M ) ) |
35 |
33 34
|
sylib |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) /\ r e. M ) ) |
36 |
35
|
simpld |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) ) |
37 |
|
xp1st |
|- ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) -> ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) ) |
38 |
36 37
|
syl |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) ) |
39 |
|
elrelimasn |
|- ( Rel V -> ( ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) <-> ( 1st ` z ) V ( 1st ` r ) ) ) |
40 |
39
|
biimpa |
|- ( ( Rel V /\ ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) ) -> ( 1st ` z ) V ( 1st ` r ) ) |
41 |
32 38 40
|
syl2anc |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` z ) V ( 1st ` r ) ) |
42 |
|
simp-4r |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> M C_ ( X X. X ) ) |
43 |
|
xpss |
|- ( X X. X ) C_ ( _V X. _V ) |
44 |
42 43
|
sstrdi |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> M C_ ( _V X. _V ) ) |
45 |
|
df-rel |
|- ( Rel M <-> M C_ ( _V X. _V ) ) |
46 |
44 45
|
sylibr |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> Rel M ) |
47 |
35
|
simprd |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> r e. M ) |
48 |
|
1st2ndbr |
|- ( ( Rel M /\ r e. M ) -> ( 1st ` r ) M ( 2nd ` r ) ) |
49 |
46 47 48
|
syl2anc |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` r ) M ( 2nd ` r ) ) |
50 |
|
xp2nd |
|- ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) -> ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) ) |
51 |
36 50
|
syl |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) ) |
52 |
|
elrelimasn |
|- ( Rel V -> ( ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) ) |
53 |
52
|
biimpa |
|- ( ( Rel V /\ ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) ) -> ( 2nd ` z ) V ( 2nd ` r ) ) |
54 |
32 51 53
|
syl2anc |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 2nd ` z ) V ( 2nd ` r ) ) |
55 |
|
simpr1r |
|- ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( ( V e. U /\ `' V = V ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) ) -> `' V = V ) |
56 |
55
|
3anassrs |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> `' V = V ) |
57 |
|
breq |
|- ( `' V = V -> ( ( 2nd ` r ) `' V ( 2nd ` z ) <-> ( 2nd ` r ) V ( 2nd ` z ) ) ) |
58 |
|
fvex |
|- ( 2nd ` r ) e. _V |
59 |
|
fvex |
|- ( 2nd ` z ) e. _V |
60 |
58 59
|
brcnv |
|- ( ( 2nd ` r ) `' V ( 2nd ` z ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) |
61 |
57 60
|
bitr3di |
|- ( `' V = V -> ( ( 2nd ` r ) V ( 2nd ` z ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) ) |
62 |
56 61
|
syl |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( ( 2nd ` r ) V ( 2nd ` z ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) ) |
63 |
54 62
|
mpbird |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 2nd ` r ) V ( 2nd ` z ) ) |
64 |
|
fvex |
|- ( 1st ` z ) e. _V |
65 |
|
fvex |
|- ( 1st ` r ) e. _V |
66 |
|
brcogw |
|- ( ( ( ( 1st ` z ) e. _V /\ ( 2nd ` r ) e. _V /\ ( 1st ` r ) e. _V ) /\ ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) ) |
67 |
66
|
ex |
|- ( ( ( 1st ` z ) e. _V /\ ( 2nd ` r ) e. _V /\ ( 1st ` r ) e. _V ) -> ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) ) ) |
68 |
64 58 65 67
|
mp3an |
|- ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) ) |
69 |
|
brcogw |
|- ( ( ( ( 1st ` z ) e. _V /\ ( 2nd ` z ) e. _V /\ ( 2nd ` r ) e. _V ) /\ ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) |
70 |
69
|
ex |
|- ( ( ( 1st ` z ) e. _V /\ ( 2nd ` z ) e. _V /\ ( 2nd ` r ) e. _V ) -> ( ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) ) |
71 |
64 59 58 70
|
mp3an |
|- ( ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) |
72 |
68 71
|
sylan |
|- ( ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) |
73 |
41 49 63 72
|
syl21anc |
|- ( ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) |
74 |
73
|
ralrimiva |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> A. r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) |
75 |
|
simplll |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> U e. ( UnifOn ` X ) ) |
76 |
|
simplrl |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> V e. U ) |
77 |
4
|
3ad2ant1 |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> J e. Top ) |
78 |
|
xp1st |
|- ( z e. ( X X. X ) -> ( 1st ` z ) e. X ) |
79 |
1
|
utopsnnei |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ ( 1st ` z ) e. X ) -> ( V " { ( 1st ` z ) } ) e. ( ( nei ` J ) ` { ( 1st ` z ) } ) ) |
80 |
78 79
|
syl3an3 |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( V " { ( 1st ` z ) } ) e. ( ( nei ` J ) ` { ( 1st ` z ) } ) ) |
81 |
|
xp2nd |
|- ( z e. ( X X. X ) -> ( 2nd ` z ) e. X ) |
82 |
1
|
utopsnnei |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ ( 2nd ` z ) e. X ) -> ( V " { ( 2nd ` z ) } ) e. ( ( nei ` J ) ` { ( 2nd ` z ) } ) ) |
83 |
81 82
|
syl3an3 |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( V " { ( 2nd ` z ) } ) e. ( ( nei ` J ) ` { ( 2nd ` z ) } ) ) |
84 |
14 14
|
neitx |
|- ( ( ( J e. Top /\ J e. Top ) /\ ( ( V " { ( 1st ` z ) } ) e. ( ( nei ` J ) ` { ( 1st ` z ) } ) /\ ( V " { ( 2nd ` z ) } ) e. ( ( nei ` J ) ` { ( 2nd ` z ) } ) ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) ) |
85 |
77 77 80 83 84
|
syl22anc |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) ) |
86 |
|
1st2nd2 |
|- ( z e. ( X X. X ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. ) |
87 |
86
|
sneqd |
|- ( z e. ( X X. X ) -> { z } = { <. ( 1st ` z ) , ( 2nd ` z ) >. } ) |
88 |
64 59
|
xpsn |
|- ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) = { <. ( 1st ` z ) , ( 2nd ` z ) >. } |
89 |
87 88
|
eqtr4di |
|- ( z e. ( X X. X ) -> { z } = ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) |
90 |
89
|
fveq2d |
|- ( z e. ( X X. X ) -> ( ( nei ` ( J tX J ) ) ` { z } ) = ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) ) |
91 |
90
|
3ad2ant3 |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( ( nei ` ( J tX J ) ) ` { z } ) = ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) ) |
92 |
85 91
|
eleqtrrd |
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { z } ) ) |
93 |
75 76 25 92
|
syl3anc |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { z } ) ) |
94 |
20
|
neindisj |
|- ( ( ( ( J tX J ) e. Top /\ M C_ U. ( J tX J ) ) /\ ( z e. ( ( cls ` ( J tX J ) ) ` M ) /\ ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { z } ) ) ) -> ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) =/= (/) ) |
95 |
7 19 24 93 94
|
syl22anc |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) =/= (/) ) |
96 |
|
r19.3rzv |
|- ( ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) =/= (/) -> ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> A. r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) ) |
97 |
95 96
|
syl |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> A. r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) ) |
98 |
74 97
|
mpbird |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) |
99 |
|
df-br |
|- ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( V o. ( M o. V ) ) ) |
100 |
98 99
|
sylib |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( V o. ( M o. V ) ) ) |
101 |
27 100
|
eqeltrd |
|- ( ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z e. ( V o. ( M o. V ) ) ) |
102 |
101
|
ex |
|- ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) -> ( z e. ( ( cls ` ( J tX J ) ) ` M ) -> z e. ( V o. ( M o. V ) ) ) ) |
103 |
102
|
ssrdv |
|- ( ( ( U e. ( UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ ( V o. ( M o. V ) ) ) |