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Theorem opnfbas 17537
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 3258 . . . 4  |-  { x  e.  J  |  S  C_  x }  C_  J
2 opnfbas.1 . . . . . 6  |-  X  = 
U. J
32eqimss2i 3233 . . . . 5  |-  U. J  C_  X
4 sspwuni 3987 . . . . 5  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
53, 4mpbir 200 . . . 4  |-  J  C_  ~P X
61, 5sstri 3188 . . 3  |-  { x  e.  J  |  S  C_  x }  C_  ~P X
76a1i 10 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  C_  ~P X )
82topopn 16652 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
98anim1i 551 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  e.  J  /\  S  C_  X ) )
1093adant3 975 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X  e.  J  /\  S  C_  X ) )
11 sseq2 3200 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1211elrab 2923 . . . . 5  |-  ( X  e.  { x  e.  J  |  S  C_  x }  <->  ( X  e.  J  /\  S  C_  X ) )
1310, 12sylibr 203 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  { x  e.  J  |  S  C_  x }
)
14 ne0i 3461 . . . 4  |-  ( X  e.  { x  e.  J  |  S  C_  x }  ->  { x  e.  J  |  S  C_  x }  =/=  (/) )
1513, 14syl 15 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  =/=  (/) )
16 ss0 3485 . . . . . . 7  |-  ( S 
C_  (/)  ->  S  =  (/) )
1716necon3ai 2486 . . . . . 6  |-  ( S  =/=  (/)  ->  -.  S  C_  (/) )
18173ad2ant3 978 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  S  C_  (/) )
1918intnand 882 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
20 df-nel 2449 . . . . 5  |-  ( (/)  e/ 
{ x  e.  J  |  S  C_  x }  <->  -.  (/)  e.  { x  e.  J  |  S  C_  x } )
21 sseq2 3200 . . . . . . 7  |-  ( x  =  (/)  ->  ( S 
C_  x  <->  S  C_  (/) ) )
2221elrab 2923 . . . . . 6  |-  ( (/)  e.  { x  e.  J  |  S  C_  x }  <->  (
(/)  e.  J  /\  S  C_  (/) ) )
2322notbii 287 . . . . 5  |-  ( -.  (/)  e.  { x  e.  J  |  S  C_  x }  <->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
2420, 23bitr2i 241 . . . 4  |-  ( -.  ( (/)  e.  J  /\  S  C_  (/) )  <->  (/)  e/  {
x  e.  J  |  S  C_  x } )
2519, 24sylib 188 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (/)  e/  {
x  e.  J  |  S  C_  x } )
26 sseq2 3200 . . . . . . 7  |-  ( x  =  r  ->  ( S  C_  x  <->  S  C_  r
) )
2726elrab 2923 . . . . . 6  |-  ( r  e.  { x  e.  J  |  S  C_  x }  <->  ( r  e.  J  /\  S  C_  r ) )
28 sseq2 3200 . . . . . . 7  |-  ( x  =  s  ->  ( S  C_  x  <->  S  C_  s
) )
2928elrab 2923 . . . . . 6  |-  ( s  e.  { x  e.  J  |  S  C_  x }  <->  ( s  e.  J  /\  S  C_  s ) )
3027, 29anbi12i 678 . . . . 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 443 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  J  e.  Top )
32 simprll 738 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  r  e.  J )
33 simprrl 740 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  s  e.  J )
34 inopn 16645 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  r  e.  J  /\  s  e.  J )  ->  ( r  i^i  s
)  e.  J )
3531, 32, 33, 34syl3anc 1182 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  ( r  i^i  s )  e.  J
)
36 ssin 3391 . . . . . . . . . . . . 13  |-  ( ( S  C_  r  /\  S  C_  s )  <->  S  C_  (
r  i^i  s )
)
3736biimpi 186 . . . . . . . . . . . 12  |-  ( ( S  C_  r  /\  S  C_  s )  ->  S  C_  ( r  i^i  s ) )
3837ad2ant2l 726 . . . . . . . . . . 11  |-  ( ( ( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) )  ->  S  C_  ( r  i^i  s ) )
3938adantl 452 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  S  C_  (
r  i^i  s )
)
4035, 39jca 518 . . . . . . . . 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 1117 . . . . . . . 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 3200 . . . . . . . . 9  |-  ( x  =  ( r  i^i  s )  ->  ( S  C_  x  <->  S  C_  (
r  i^i  s )
) )
4342elrab 2923 . . . . . . . 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 203 . . . . . . 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 3197 . . . . . . 7  |-  ( r  i^i  s )  C_  ( r  i^i  s
)
46 sseq1 3199 . . . . . . . 8  |-  ( t  =  ( r  i^i  s )  ->  (
t  C_  ( r  i^i  s )  <->  ( r  i^i  s )  C_  (
r  i^i  s )
) )
4746rspcev 2884 . . . . . . 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 643 . . . . . 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 423 . . . . 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 208 . . . 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 2634 . . 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 1132 . 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 17530 . . . 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 15 . . 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 976 . 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 888 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  { x  e.  J  |  S  C_  x }  e.  (
fBas `  X )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   ` cfv 5255   Topctop 16631   fBascfbas 17518
This theorem is referenced by:  neifg  26320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-fbas 17520
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