MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opnfbas Structured version   Unicode version

Theorem opnfbas 19542
Description: The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
Hypothesis
Ref Expression
opnfbas.1  |-  X  = 
U. J
Assertion
Ref Expression
opnfbas  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
Distinct variable groups:    x, J    x, S    x, X

Proof of Theorem opnfbas
Dummy variables  s 
r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3540 . . . 4  |-  { x  e.  J  |  S  C_  x }  C_  J
2 opnfbas.1 . . . . . 6  |-  X  = 
U. J
32eqimss2i 3514 . . . . 5  |-  U. J  C_  X
4 sspwuni 4359 . . . . 5  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
53, 4mpbir 209 . . . 4  |-  J  C_  ~P X
61, 5sstri 3468 . . 3  |-  { x  e.  J  |  S  C_  x }  C_  ~P X
76a1i 11 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  C_  ~P X )
82topopn 18646 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
98anim1i 568 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  e.  J  /\  S  C_  X ) )
1093adant3 1008 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X  e.  J  /\  S  C_  X ) )
11 sseq2 3481 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1211elrab 3218 . . . . 5  |-  ( X  e.  { x  e.  J  |  S  C_  x }  <->  ( X  e.  J  /\  S  C_  X ) )
1310, 12sylibr 212 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  { x  e.  J  |  S  C_  x }
)
14 ne0i 3746 . . . 4  |-  ( X  e.  { x  e.  J  |  S  C_  x }  ->  { x  e.  J  |  S  C_  x }  =/=  (/) )
1513, 14syl 16 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  =/=  (/) )
16 ss0 3771 . . . . . . 7  |-  ( S 
C_  (/)  ->  S  =  (/) )
1716necon3ai 2677 . . . . . 6  |-  ( S  =/=  (/)  ->  -.  S  C_  (/) )
18173ad2ant3 1011 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  S  C_  (/) )
1918intnand 907 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
20 df-nel 2648 . . . . 5  |-  ( (/)  e/ 
{ x  e.  J  |  S  C_  x }  <->  -.  (/)  e.  { x  e.  J  |  S  C_  x } )
21 sseq2 3481 . . . . . . 7  |-  ( x  =  (/)  ->  ( S 
C_  x  <->  S  C_  (/) ) )
2221elrab 3218 . . . . . 6  |-  ( (/)  e.  { x  e.  J  |  S  C_  x }  <->  (
(/)  e.  J  /\  S  C_  (/) ) )
2322notbii 296 . . . . 5  |-  ( -.  (/)  e.  { x  e.  J  |  S  C_  x }  <->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
2420, 23bitr2i 250 . . . 4  |-  ( -.  ( (/)  e.  J  /\  S  C_  (/) )  <->  (/)  e/  {
x  e.  J  |  S  C_  x } )
2519, 24sylib 196 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (/)  e/  {
x  e.  J  |  S  C_  x } )
26 sseq2 3481 . . . . . . 7  |-  ( x  =  r  ->  ( S  C_  x  <->  S  C_  r
) )
2726elrab 3218 . . . . . 6  |-  ( r  e.  { x  e.  J  |  S  C_  x }  <->  ( r  e.  J  /\  S  C_  r ) )
28 sseq2 3481 . . . . . . 7  |-  ( x  =  s  ->  ( S  C_  x  <->  S  C_  s
) )
2928elrab 3218 . . . . . 6  |-  ( s  e.  { x  e.  J  |  S  C_  x }  <->  ( s  e.  J  /\  S  C_  s ) )
3027, 29anbi12i 697 . . . . 5  |-  ( ( r  e.  { x  e.  J  |  S  C_  x }  /\  s  e.  { x  e.  J  |  S  C_  x }
)  <->  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )
31 simpl 457 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  J  e.  Top )
32 simprll 761 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  r  e.  J )
33 simprrl 763 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  s  e.  J )
34 inopn 18639 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  r  e.  J  /\  s  e.  J )  ->  ( r  i^i  s
)  e.  J )
3531, 32, 33, 34syl3anc 1219 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  ( r  i^i  s )  e.  J
)
36 ssin 3675 . . . . . . . . . . . . 13  |-  ( ( S  C_  r  /\  S  C_  s )  <->  S  C_  (
r  i^i  s )
)
3736biimpi 194 . . . . . . . . . . . 12  |-  ( ( S  C_  r  /\  S  C_  s )  ->  S  C_  ( r  i^i  s ) )
3837ad2ant2l 745 . . . . . . . . . . 11  |-  ( ( ( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) )  ->  S  C_  ( r  i^i  s ) )
3938adantl 466 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  S  C_  (
r  i^i  s )
)
4035, 39jca 532 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  ( (
r  i^i  s )  e.  J  /\  S  C_  ( r  i^i  s
) ) )
41403ad2antl1 1150 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  /\  (
( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) ) )  ->  ( ( r  i^i  s )  e.  J  /\  S  C_  ( r  i^i  s
) ) )
42 sseq2 3481 . . . . . . . . 9  |-  ( x  =  ( r  i^i  s )  ->  ( S  C_  x  <->  S  C_  (
r  i^i  s )
) )
4342elrab 3218 . . . . . . . 8  |-  ( ( r  i^i  s )  e.  { x  e.  J  |  S  C_  x }  <->  ( ( r  i^i  s )  e.  J  /\  S  C_  ( r  i^i  s
) ) )
4441, 43sylibr 212 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  /\  (
( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) ) )  ->  ( r  i^i  s )  e.  {
x  e.  J  |  S  C_  x } )
45 ssid 3478 . . . . . . 7  |-  ( r  i^i  s )  C_  ( r  i^i  s
)
46 sseq1 3480 . . . . . . . 8  |-  ( t  =  ( r  i^i  s )  ->  (
t  C_  ( r  i^i  s )  <->  ( r  i^i  s )  C_  (
r  i^i  s )
) )
4746rspcev 3173 . . . . . . 7  |-  ( ( ( r  i^i  s
)  e.  { x  e.  J  |  S  C_  x }  /\  (
r  i^i  s )  C_  ( r  i^i  s
) )  ->  E. t  e.  { x  e.  J  |  S  C_  x }
t  C_  ( r  i^i  s ) )
4844, 45, 47sylancl 662 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  /\  (
( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) ) )  ->  E. t  e.  {
x  e.  J  |  S  C_  x } t 
C_  ( r  i^i  s ) )
4948ex 434 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) )  ->  E. t  e.  {
x  e.  J  |  S  C_  x } t 
C_  ( r  i^i  s ) ) )
5030, 49syl5bi 217 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (
( r  e.  {
x  e.  J  |  S  C_  x }  /\  s  e.  { x  e.  J  |  S  C_  x } )  ->  E. t  e.  { x  e.  J  |  S  C_  x } t  C_  ( r  i^i  s
) ) )
5150ralrimivv 2907 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  A. r  e.  { x  e.  J  |  S  C_  x } A. s  e.  { x  e.  J  |  S  C_  x } E. t  e.  { x  e.  J  |  S  C_  x }
t  C_  ( r  i^i  s ) )
5215, 25, 513jca 1168 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( { x  e.  J  |  S  C_  x }  =/=  (/)  /\  (/)  e/  {
x  e.  J  |  S  C_  x }  /\  A. r  e.  { x  e.  J  |  S  C_  x } A. s  e.  { x  e.  J  |  S  C_  x } E. t  e.  { x  e.  J  |  S  C_  x } t  C_  ( r  i^i  s
) ) )
53 isfbas2 19535 . . . 4  |-  ( X  e.  J  ->  ( { x  e.  J  |  S  C_  x }  e.  ( fBas `  X
)  <->  ( { x  e.  J  |  S  C_  x }  C_  ~P X  /\  ( { x  e.  J  |  S  C_  x }  =/=  (/)  /\  (/)  e/  {
x  e.  J  |  S  C_  x }  /\  A. r  e.  { x  e.  J  |  S  C_  x } A. s  e.  { x  e.  J  |  S  C_  x } E. t  e.  { x  e.  J  |  S  C_  x } t  C_  ( r  i^i  s
) ) ) ) )
548, 53syl 16 . . 3  |-  ( J  e.  Top  ->  ( { x  e.  J  |  S  C_  x }  e.  ( fBas `  X
)  <->  ( { x  e.  J  |  S  C_  x }  C_  ~P X  /\  ( { x  e.  J  |  S  C_  x }  =/=  (/)  /\  (/)  e/  {
x  e.  J  |  S  C_  x }  /\  A. r  e.  { x  e.  J  |  S  C_  x } A. s  e.  { x  e.  J  |  S  C_  x } E. t  e.  { x  e.  J  |  S  C_  x } t  C_  ( r  i^i  s
) ) ) ) )
55543ad2ant1 1009 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( { x  e.  J  |  S  C_  x }  e.  ( fBas `  X
)  <->  ( { x  e.  J  |  S  C_  x }  C_  ~P X  /\  ( { x  e.  J  |  S  C_  x }  =/=  (/)  /\  (/)  e/  {
x  e.  J  |  S  C_  x }  /\  A. r  e.  { x  e.  J  |  S  C_  x } A. s  e.  { x  e.  J  |  S  C_  x } E. t  e.  { x  e.  J  |  S  C_  x } t  C_  ( r  i^i  s
) ) ) ) )
567, 52, 55mpbir2and 913 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645    e/ wnel 2646   A.wral 2796   E.wrex 2797   {crab 2800    i^i cin 3430    C_ wss 3431   (/)c0 3740   ~Pcpw 3963   U.cuni 4194   ` cfv 5521   fBascfbas 17924   Topctop 18625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fv 5529  df-fbas 17934  df-top 18630
This theorem is referenced by:  neifg  28735
  Copyright terms: Public domain W3C validator