Step |
Hyp |
Ref |
Expression |
1 |
|
2ndcctbss.1 |
|- J = ( topGen ` B ) |
2 |
|
2ndcctbss.2 |
|- S = { <. u , v >. | ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) } |
3 |
|
simpr |
|- ( ( B e. TopBases /\ J e. 2ndc ) -> J e. 2ndc ) |
4 |
|
is2ndc |
|- ( J e. 2ndc <-> E. c e. TopBases ( c ~<_ _om /\ ( topGen ` c ) = J ) ) |
5 |
3 4
|
sylib |
|- ( ( B e. TopBases /\ J e. 2ndc ) -> E. c e. TopBases ( c ~<_ _om /\ ( topGen ` c ) = J ) ) |
6 |
|
vex |
|- c e. _V |
7 |
6 6
|
xpex |
|- ( c X. c ) e. _V |
8 |
|
3simpa |
|- ( ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) -> ( u e. c /\ v e. c ) ) |
9 |
8
|
ssopab2i |
|- { <. u , v >. | ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) } C_ { <. u , v >. | ( u e. c /\ v e. c ) } |
10 |
|
df-xp |
|- ( c X. c ) = { <. u , v >. | ( u e. c /\ v e. c ) } |
11 |
9 2 10
|
3sstr4i |
|- S C_ ( c X. c ) |
12 |
|
ssdomg |
|- ( ( c X. c ) e. _V -> ( S C_ ( c X. c ) -> S ~<_ ( c X. c ) ) ) |
13 |
7 11 12
|
mp2 |
|- S ~<_ ( c X. c ) |
14 |
6
|
xpdom1 |
|- ( c ~<_ _om -> ( c X. c ) ~<_ ( _om X. c ) ) |
15 |
|
omex |
|- _om e. _V |
16 |
15
|
xpdom2 |
|- ( c ~<_ _om -> ( _om X. c ) ~<_ ( _om X. _om ) ) |
17 |
|
domtr |
|- ( ( ( c X. c ) ~<_ ( _om X. c ) /\ ( _om X. c ) ~<_ ( _om X. _om ) ) -> ( c X. c ) ~<_ ( _om X. _om ) ) |
18 |
14 16 17
|
syl2anc |
|- ( c ~<_ _om -> ( c X. c ) ~<_ ( _om X. _om ) ) |
19 |
|
xpomen |
|- ( _om X. _om ) ~~ _om |
20 |
|
domentr |
|- ( ( ( c X. c ) ~<_ ( _om X. _om ) /\ ( _om X. _om ) ~~ _om ) -> ( c X. c ) ~<_ _om ) |
21 |
18 19 20
|
sylancl |
|- ( c ~<_ _om -> ( c X. c ) ~<_ _om ) |
22 |
21
|
adantr |
|- ( ( c ~<_ _om /\ ( topGen ` c ) = J ) -> ( c X. c ) ~<_ _om ) |
23 |
22
|
ad2antll |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( c X. c ) ~<_ _om ) |
24 |
|
domtr |
|- ( ( S ~<_ ( c X. c ) /\ ( c X. c ) ~<_ _om ) -> S ~<_ _om ) |
25 |
13 23 24
|
sylancr |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> S ~<_ _om ) |
26 |
2
|
relopabiv |
|- Rel S |
27 |
|
simpr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> x e. S ) |
28 |
|
1st2nd |
|- ( ( Rel S /\ x e. S ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
29 |
26 27 28
|
sylancr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
30 |
29 27
|
eqeltrrd |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. S ) |
31 |
|
df-br |
|- ( ( 1st ` x ) S ( 2nd ` x ) <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. S ) |
32 |
|
fvex |
|- ( 1st ` x ) e. _V |
33 |
|
fvex |
|- ( 2nd ` x ) e. _V |
34 |
|
simpl |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> u = ( 1st ` x ) ) |
35 |
34
|
eleq1d |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( u e. c <-> ( 1st ` x ) e. c ) ) |
36 |
|
simpr |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> v = ( 2nd ` x ) ) |
37 |
36
|
eleq1d |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( v e. c <-> ( 2nd ` x ) e. c ) ) |
38 |
|
sseq1 |
|- ( u = ( 1st ` x ) -> ( u C_ w <-> ( 1st ` x ) C_ w ) ) |
39 |
|
sseq2 |
|- ( v = ( 2nd ` x ) -> ( w C_ v <-> w C_ ( 2nd ` x ) ) ) |
40 |
38 39
|
bi2anan9 |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( ( u C_ w /\ w C_ v ) <-> ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) ) |
41 |
40
|
rexbidv |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( E. w e. B ( u C_ w /\ w C_ v ) <-> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) ) |
42 |
35 37 41
|
3anbi123d |
|- ( ( u = ( 1st ` x ) /\ v = ( 2nd ` x ) ) -> ( ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) <-> ( ( 1st ` x ) e. c /\ ( 2nd ` x ) e. c /\ E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) ) ) |
43 |
32 33 42 2
|
braba |
|- ( ( 1st ` x ) S ( 2nd ` x ) <-> ( ( 1st ` x ) e. c /\ ( 2nd ` x ) e. c /\ E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) ) |
44 |
31 43
|
bitr3i |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. S <-> ( ( 1st ` x ) e. c /\ ( 2nd ` x ) e. c /\ E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) ) |
45 |
44
|
simp3bi |
|- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. S -> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) |
46 |
30 45
|
syl |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> E. w e. B ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) |
47 |
|
fvi |
|- ( B e. TopBases -> ( _I ` B ) = B ) |
48 |
47
|
ad3antrrr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> ( _I ` B ) = B ) |
49 |
46 48
|
rexeqtrrdv |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ x e. S ) -> E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) |
50 |
49
|
ralrimiva |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> A. x e. S E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) |
51 |
|
fvex |
|- ( _I ` B ) e. _V |
52 |
|
sseq2 |
|- ( w = ( f ` x ) -> ( ( 1st ` x ) C_ w <-> ( 1st ` x ) C_ ( f ` x ) ) ) |
53 |
|
sseq1 |
|- ( w = ( f ` x ) -> ( w C_ ( 2nd ` x ) <-> ( f ` x ) C_ ( 2nd ` x ) ) ) |
54 |
52 53
|
anbi12d |
|- ( w = ( f ` x ) -> ( ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) <-> ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) |
55 |
51 54
|
axcc4dom |
|- ( ( S ~<_ _om /\ A. x e. S E. w e. ( _I ` B ) ( ( 1st ` x ) C_ w /\ w C_ ( 2nd ` x ) ) ) -> E. f ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) |
56 |
25 50 55
|
syl2anc |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> E. f ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) |
57 |
47
|
ad2antrr |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( _I ` B ) = B ) |
58 |
57
|
feq3d |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( f : S --> ( _I ` B ) <-> f : S --> B ) ) |
59 |
58
|
anbi1d |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) <-> ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) ) |
60 |
|
2ndctop |
|- ( J e. 2ndc -> J e. Top ) |
61 |
60
|
adantl |
|- ( ( B e. TopBases /\ J e. 2ndc ) -> J e. Top ) |
62 |
61
|
ad2antrr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> J e. Top ) |
63 |
|
frn |
|- ( f : S --> B -> ran f C_ B ) |
64 |
63
|
ad2antrl |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f C_ B ) |
65 |
|
bastg |
|- ( B e. TopBases -> B C_ ( topGen ` B ) ) |
66 |
65
|
ad3antrrr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> B C_ ( topGen ` B ) ) |
67 |
66 1
|
sseqtrrdi |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> B C_ J ) |
68 |
64 67
|
sstrd |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f C_ J ) |
69 |
|
simprrl |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> o e. J ) |
70 |
|
simprr |
|- ( ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) -> ( topGen ` c ) = J ) |
71 |
70
|
ad2antlr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> ( topGen ` c ) = J ) |
72 |
69 71
|
eleqtrrd |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> o e. ( topGen ` c ) ) |
73 |
|
simprrr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> t e. o ) |
74 |
|
tg2 |
|- ( ( o e. ( topGen ` c ) /\ t e. o ) -> E. d e. c ( t e. d /\ d C_ o ) ) |
75 |
72 73 74
|
syl2anc |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> E. d e. c ( t e. d /\ d C_ o ) ) |
76 |
|
bastg |
|- ( c e. TopBases -> c C_ ( topGen ` c ) ) |
77 |
76
|
ad2antrl |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> c C_ ( topGen ` c ) ) |
78 |
77
|
ad2antrr |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> c C_ ( topGen ` c ) ) |
79 |
1
|
eqeq2i |
|- ( ( topGen ` c ) = J <-> ( topGen ` c ) = ( topGen ` B ) ) |
80 |
79
|
biimpi |
|- ( ( topGen ` c ) = J -> ( topGen ` c ) = ( topGen ` B ) ) |
81 |
80
|
adantl |
|- ( ( c ~<_ _om /\ ( topGen ` c ) = J ) -> ( topGen ` c ) = ( topGen ` B ) ) |
82 |
81
|
ad2antll |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( topGen ` c ) = ( topGen ` B ) ) |
83 |
82
|
ad2antrr |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> ( topGen ` c ) = ( topGen ` B ) ) |
84 |
78 83
|
sseqtrd |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> c C_ ( topGen ` B ) ) |
85 |
|
simprl |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> d e. c ) |
86 |
84 85
|
sseldd |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> d e. ( topGen ` B ) ) |
87 |
|
simprrl |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> t e. d ) |
88 |
|
tg2 |
|- ( ( d e. ( topGen ` B ) /\ t e. d ) -> E. m e. B ( t e. m /\ m C_ d ) ) |
89 |
86 87 88
|
syl2anc |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> E. m e. B ( t e. m /\ m C_ d ) ) |
90 |
65
|
ad3antrrr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> B C_ ( topGen ` B ) ) |
91 |
90
|
ad2antrr |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> B C_ ( topGen ` B ) ) |
92 |
71
|
ad2antrr |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> ( topGen ` c ) = J ) |
93 |
92 1
|
eqtr2di |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> ( topGen ` B ) = ( topGen ` c ) ) |
94 |
91 93
|
sseqtrd |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> B C_ ( topGen ` c ) ) |
95 |
|
simprl |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> m e. B ) |
96 |
94 95
|
sseldd |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> m e. ( topGen ` c ) ) |
97 |
|
simprrl |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> t e. m ) |
98 |
|
tg2 |
|- ( ( m e. ( topGen ` c ) /\ t e. m ) -> E. n e. c ( t e. n /\ n C_ m ) ) |
99 |
96 97 98
|
syl2anc |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> E. n e. c ( t e. n /\ n C_ m ) ) |
100 |
|
ffn |
|- ( f : S --> B -> f Fn S ) |
101 |
100
|
ad2antrr |
|- ( ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) -> f Fn S ) |
102 |
101
|
ad2antlr |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> f Fn S ) |
103 |
102
|
ad2antrr |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> f Fn S ) |
104 |
|
simprl |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n e. c ) |
105 |
85
|
ad2antrr |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> d e. c ) |
106 |
|
simplrl |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m e. B ) |
107 |
|
simprrr |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n C_ m ) |
108 |
|
simprr |
|- ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> m C_ d ) |
109 |
108
|
ad2antlr |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m C_ d ) |
110 |
|
sseq2 |
|- ( w = m -> ( n C_ w <-> n C_ m ) ) |
111 |
|
sseq1 |
|- ( w = m -> ( w C_ d <-> m C_ d ) ) |
112 |
110 111
|
anbi12d |
|- ( w = m -> ( ( n C_ w /\ w C_ d ) <-> ( n C_ m /\ m C_ d ) ) ) |
113 |
112
|
rspcev |
|- ( ( m e. B /\ ( n C_ m /\ m C_ d ) ) -> E. w e. B ( n C_ w /\ w C_ d ) ) |
114 |
106 107 109 113
|
syl12anc |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> E. w e. B ( n C_ w /\ w C_ d ) ) |
115 |
|
df-br |
|- ( n S d <-> <. n , d >. e. S ) |
116 |
|
vex |
|- n e. _V |
117 |
|
vex |
|- d e. _V |
118 |
|
simpl |
|- ( ( u = n /\ v = d ) -> u = n ) |
119 |
118
|
eleq1d |
|- ( ( u = n /\ v = d ) -> ( u e. c <-> n e. c ) ) |
120 |
|
simpr |
|- ( ( u = n /\ v = d ) -> v = d ) |
121 |
120
|
eleq1d |
|- ( ( u = n /\ v = d ) -> ( v e. c <-> d e. c ) ) |
122 |
|
sseq1 |
|- ( u = n -> ( u C_ w <-> n C_ w ) ) |
123 |
|
sseq2 |
|- ( v = d -> ( w C_ v <-> w C_ d ) ) |
124 |
122 123
|
bi2anan9 |
|- ( ( u = n /\ v = d ) -> ( ( u C_ w /\ w C_ v ) <-> ( n C_ w /\ w C_ d ) ) ) |
125 |
124
|
rexbidv |
|- ( ( u = n /\ v = d ) -> ( E. w e. B ( u C_ w /\ w C_ v ) <-> E. w e. B ( n C_ w /\ w C_ d ) ) ) |
126 |
119 121 125
|
3anbi123d |
|- ( ( u = n /\ v = d ) -> ( ( u e. c /\ v e. c /\ E. w e. B ( u C_ w /\ w C_ v ) ) <-> ( n e. c /\ d e. c /\ E. w e. B ( n C_ w /\ w C_ d ) ) ) ) |
127 |
116 117 126 2
|
braba |
|- ( n S d <-> ( n e. c /\ d e. c /\ E. w e. B ( n C_ w /\ w C_ d ) ) ) |
128 |
115 127
|
bitr3i |
|- ( <. n , d >. e. S <-> ( n e. c /\ d e. c /\ E. w e. B ( n C_ w /\ w C_ d ) ) ) |
129 |
104 105 114 128
|
syl3anbrc |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> <. n , d >. e. S ) |
130 |
|
fnfvelrn |
|- ( ( f Fn S /\ <. n , d >. e. S ) -> ( f ` <. n , d >. ) e. ran f ) |
131 |
103 129 130
|
syl2anc |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( f ` <. n , d >. ) e. ran f ) |
132 |
|
simprl |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n e. c ) |
133 |
|
simplll |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> d e. c ) |
134 |
|
simplrl |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m e. B ) |
135 |
|
simprrr |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> n C_ m ) |
136 |
108
|
ad2antlr |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> m C_ d ) |
137 |
134 135 136 113
|
syl12anc |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> E. w e. B ( n C_ w /\ w C_ d ) ) |
138 |
132 133 137 128
|
syl3anbrc |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> <. n , d >. e. S ) |
139 |
|
fveq2 |
|- ( x = <. n , d >. -> ( 1st ` x ) = ( 1st ` <. n , d >. ) ) |
140 |
|
fveq2 |
|- ( x = <. n , d >. -> ( f ` x ) = ( f ` <. n , d >. ) ) |
141 |
139 140
|
sseq12d |
|- ( x = <. n , d >. -> ( ( 1st ` x ) C_ ( f ` x ) <-> ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) ) ) |
142 |
|
fveq2 |
|- ( x = <. n , d >. -> ( 2nd ` x ) = ( 2nd ` <. n , d >. ) ) |
143 |
140 142
|
sseq12d |
|- ( x = <. n , d >. -> ( ( f ` x ) C_ ( 2nd ` x ) <-> ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) |
144 |
141 143
|
anbi12d |
|- ( x = <. n , d >. -> ( ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) <-> ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) ) |
145 |
144
|
rspcv |
|- ( <. n , d >. e. S -> ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) ) |
146 |
138 145
|
syl |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) ) ) |
147 |
116 117
|
op1st |
|- ( 1st ` <. n , d >. ) = n |
148 |
147
|
sseq1i |
|- ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) <-> n C_ ( f ` <. n , d >. ) ) |
149 |
116 117
|
op2nd |
|- ( 2nd ` <. n , d >. ) = d |
150 |
149
|
sseq2i |
|- ( ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) <-> ( f ` <. n , d >. ) C_ d ) |
151 |
148 150
|
anbi12i |
|- ( ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) <-> ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) |
152 |
|
simprl |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> n C_ ( f ` <. n , d >. ) ) |
153 |
|
simprl |
|- ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> t e. n ) |
154 |
153
|
ad2antlr |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> t e. n ) |
155 |
152 154
|
sseldd |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> t e. ( f ` <. n , d >. ) ) |
156 |
|
simprr |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> ( f ` <. n , d >. ) C_ d ) |
157 |
|
simplrr |
|- ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> d C_ o ) |
158 |
157
|
ad2antrr |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> d C_ o ) |
159 |
156 158
|
sstrd |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> ( f ` <. n , d >. ) C_ o ) |
160 |
155 159
|
jca |
|- ( ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) /\ ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) |
161 |
160
|
ex |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( ( n C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ d ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) |
162 |
151 161
|
biimtrid |
|- ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( ( ( 1st ` <. n , d >. ) C_ ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ ( 2nd ` <. n , d >. ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) |
163 |
146 162
|
syldc |
|- ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( ( ( d e. c /\ ( t e. d /\ d C_ o ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) |
164 |
163
|
exp4c |
|- ( A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) -> ( ( d e. c /\ ( t e. d /\ d C_ o ) ) -> ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) ) ) |
165 |
164
|
ad2antlr |
|- ( ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) -> ( ( d e. c /\ ( t e. d /\ d C_ o ) ) -> ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) ) ) |
166 |
165
|
adantl |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> ( ( d e. c /\ ( t e. d /\ d C_ o ) ) -> ( ( m e. B /\ ( t e. m /\ m C_ d ) ) -> ( ( n e. c /\ ( t e. n /\ n C_ m ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) ) ) |
167 |
166
|
imp41 |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) |
168 |
|
eleq2 |
|- ( b = ( f ` <. n , d >. ) -> ( t e. b <-> t e. ( f ` <. n , d >. ) ) ) |
169 |
|
sseq1 |
|- ( b = ( f ` <. n , d >. ) -> ( b C_ o <-> ( f ` <. n , d >. ) C_ o ) ) |
170 |
168 169
|
anbi12d |
|- ( b = ( f ` <. n , d >. ) -> ( ( t e. b /\ b C_ o ) <-> ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) ) |
171 |
170
|
rspcev |
|- ( ( ( f ` <. n , d >. ) e. ran f /\ ( t e. ( f ` <. n , d >. ) /\ ( f ` <. n , d >. ) C_ o ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) |
172 |
131 167 171
|
syl2anc |
|- ( ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) /\ ( n e. c /\ ( t e. n /\ n C_ m ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) |
173 |
99 172
|
rexlimddv |
|- ( ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) /\ ( m e. B /\ ( t e. m /\ m C_ d ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) |
174 |
89 173
|
rexlimddv |
|- ( ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) /\ ( d e. c /\ ( t e. d /\ d C_ o ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) |
175 |
75 174
|
rexlimddv |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) /\ ( o e. J /\ t e. o ) ) ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) |
176 |
175
|
expr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ( ( o e. J /\ t e. o ) -> E. b e. ran f ( t e. b /\ b C_ o ) ) ) |
177 |
176
|
ralrimivv |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> A. o e. J A. t e. o E. b e. ran f ( t e. b /\ b C_ o ) ) |
178 |
|
basgen2 |
|- ( ( J e. Top /\ ran f C_ J /\ A. o e. J A. t e. o E. b e. ran f ( t e. b /\ b C_ o ) ) -> ( topGen ` ran f ) = J ) |
179 |
62 68 177 178
|
syl3anc |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ( topGen ` ran f ) = J ) |
180 |
179 62
|
eqeltrd |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ( topGen ` ran f ) e. Top ) |
181 |
|
tgclb |
|- ( ran f e. TopBases <-> ( topGen ` ran f ) e. Top ) |
182 |
180 181
|
sylibr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f e. TopBases ) |
183 |
|
omelon |
|- _om e. On |
184 |
25
|
adantr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> S ~<_ _om ) |
185 |
|
ondomen |
|- ( ( _om e. On /\ S ~<_ _om ) -> S e. dom card ) |
186 |
183 184 185
|
sylancr |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> S e. dom card ) |
187 |
100
|
ad2antrl |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> f Fn S ) |
188 |
|
dffn4 |
|- ( f Fn S <-> f : S -onto-> ran f ) |
189 |
187 188
|
sylib |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> f : S -onto-> ran f ) |
190 |
|
fodomnum |
|- ( S e. dom card -> ( f : S -onto-> ran f -> ran f ~<_ S ) ) |
191 |
186 189 190
|
sylc |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f ~<_ S ) |
192 |
|
domtr |
|- ( ( ran f ~<_ S /\ S ~<_ _om ) -> ran f ~<_ _om ) |
193 |
191 184 192
|
syl2anc |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> ran f ~<_ _om ) |
194 |
179
|
eqcomd |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> J = ( topGen ` ran f ) ) |
195 |
|
breq1 |
|- ( b = ran f -> ( b ~<_ _om <-> ran f ~<_ _om ) ) |
196 |
|
sseq1 |
|- ( b = ran f -> ( b C_ B <-> ran f C_ B ) ) |
197 |
|
fveq2 |
|- ( b = ran f -> ( topGen ` b ) = ( topGen ` ran f ) ) |
198 |
197
|
eqeq2d |
|- ( b = ran f -> ( J = ( topGen ` b ) <-> J = ( topGen ` ran f ) ) ) |
199 |
195 196 198
|
3anbi123d |
|- ( b = ran f -> ( ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) <-> ( ran f ~<_ _om /\ ran f C_ B /\ J = ( topGen ` ran f ) ) ) ) |
200 |
199
|
rspcev |
|- ( ( ran f e. TopBases /\ ( ran f ~<_ _om /\ ran f C_ B /\ J = ( topGen ` ran f ) ) ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) |
201 |
182 193 64 194 200
|
syl13anc |
|- ( ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) /\ ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) |
202 |
201
|
ex |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( ( f : S --> B /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) ) |
203 |
59 202
|
sylbid |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) ) |
204 |
203
|
exlimdv |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> ( E. f ( f : S --> ( _I ` B ) /\ A. x e. S ( ( 1st ` x ) C_ ( f ` x ) /\ ( f ` x ) C_ ( 2nd ` x ) ) ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) ) |
205 |
56 204
|
mpd |
|- ( ( ( B e. TopBases /\ J e. 2ndc ) /\ ( c e. TopBases /\ ( c ~<_ _om /\ ( topGen ` c ) = J ) ) ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) |
206 |
5 205
|
rexlimddv |
|- ( ( B e. TopBases /\ J e. 2ndc ) -> E. b e. TopBases ( b ~<_ _om /\ b C_ B /\ J = ( topGen ` b ) ) ) |