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Theorem opnfbas 20512
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 3571 . . . 4  |-  { x  e.  J  |  S  C_  x }  C_  J
2 opnfbas.1 . . . . . 6  |-  X  = 
U. J
32eqimss2i 3544 . . . . 5  |-  U. J  C_  X
4 sspwuni 4404 . . . . 5  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
53, 4mpbir 209 . . . 4  |-  J  C_  ~P X
61, 5sstri 3498 . . 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 19585 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
98anim1i 566 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  e.  J  /\  S  C_  X ) )
1093adant3 1014 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X  e.  J  /\  S  C_  X ) )
11 sseq2 3511 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1211elrab 3254 . . . . 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 3789 . . . 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 3815 . . . . . . 7  |-  ( S 
C_  (/)  ->  S  =  (/) )
1716necon3ai 2682 . . . . . 6  |-  ( S  =/=  (/)  ->  -.  S  C_  (/) )
18173ad2ant3 1017 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  S  C_  (/) )
1918intnand 914 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
20 df-nel 2652 . . . . 5  |-  ( (/)  e/ 
{ x  e.  J  |  S  C_  x }  <->  -.  (/)  e.  { x  e.  J  |  S  C_  x } )
21 sseq2 3511 . . . . . . 7  |-  ( x  =  (/)  ->  ( S 
C_  x  <->  S  C_  (/) ) )
2221elrab 3254 . . . . . 6  |-  ( (/)  e.  { x  e.  J  |  S  C_  x }  <->  (
(/)  e.  J  /\  S  C_  (/) ) )
2322notbii 294 . . . . 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 3511 . . . . . . 7  |-  ( x  =  r  ->  ( S  C_  x  <->  S  C_  r
) )
2726elrab 3254 . . . . . 6  |-  ( r  e.  { x  e.  J  |  S  C_  x }  <->  ( r  e.  J  /\  S  C_  r ) )
28 sseq2 3511 . . . . . . 7  |-  ( x  =  s  ->  ( S  C_  x  <->  S  C_  s
) )
2928elrab 3254 . . . . . 6  |-  ( s  e.  { x  e.  J  |  S  C_  x }  <->  ( s  e.  J  /\  S  C_  s ) )
3027, 29anbi12i 695 . . . . 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 455 . . . . . . . . . . 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 19578 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  r  e.  J  /\  s  e.  J )  ->  ( r  i^i  s
)  e.  J )
3531, 32, 33, 34syl3anc 1226 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  ( r  i^i  s )  e.  J
)
36 ssin 3706 . . . . . . . . . . . . 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 743 . . . . . . . . . . 11  |-  ( ( ( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) )  ->  S  C_  ( r  i^i  s ) )
3938adantl 464 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  S  C_  (
r  i^i  s )
)
4035, 39jca 530 . . . . . . . . 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 1156 . . . . . . . 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 3511 . . . . . . . . 9  |-  ( x  =  ( r  i^i  s )  ->  ( S  C_  x  <->  S  C_  (
r  i^i  s )
) )
4342elrab 3254 . . . . . . . 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 3508 . . . . . . 7  |-  ( r  i^i  s )  C_  ( r  i^i  s
)
46 sseq1 3510 . . . . . . . 8  |-  ( t  =  ( r  i^i  s )  ->  (
t  C_  ( r  i^i  s )  <->  ( r  i^i  s )  C_  (
r  i^i  s )
) )
4746rspcev 3207 . . . . . . 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 660 . . . . . 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 432 . . . . 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 2874 . . 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 1174 . 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 20505 . . . 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 1015 . 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 920 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    e/ wnel 2650   A.wral 2804   E.wrex 2805   {crab 2808    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   ` cfv 5570   fBascfbas 18604   Topctop 19564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fv 5578  df-fbas 18614  df-top 19569
This theorem is referenced by:  neifg  30432
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