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Theorem opnfbas 17866
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 3420 . . . 4  |-  { x  e.  J  |  S  C_  x }  C_  J
2 opnfbas.1 . . . . . 6  |-  X  = 
U. J
32eqimss2i 3395 . . . . 5  |-  U. J  C_  X
4 sspwuni 4168 . . . . 5  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
53, 4mpbir 201 . . . 4  |-  J  C_  ~P X
61, 5sstri 3349 . . 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 16971 . . . . . . 7  |-  ( J  e.  Top  ->  X  e.  J )
98anim1i 552 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( X  e.  J  /\  S  C_  X ) )
1093adant3 977 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  ( X  e.  J  /\  S  C_  X ) )
11 sseq2 3362 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1211elrab 3084 . . . . 5  |-  ( X  e.  { x  e.  J  |  S  C_  x }  <->  ( X  e.  J  /\  S  C_  X ) )
1310, 12sylibr 204 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  X  e.  { x  e.  J  |  S  C_  x }
)
14 ne0i 3626 . . . 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 3650 . . . . . . 7  |-  ( S 
C_  (/)  ->  S  =  (/) )
1716necon3ai 2638 . . . . . 6  |-  ( S  =/=  (/)  ->  -.  S  C_  (/) )
18173ad2ant3 980 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  S  C_  (/) )
1918intnand 883 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
20 df-nel 2601 . . . . 5  |-  ( (/)  e/ 
{ x  e.  J  |  S  C_  x }  <->  -.  (/)  e.  { x  e.  J  |  S  C_  x } )
21 sseq2 3362 . . . . . . 7  |-  ( x  =  (/)  ->  ( S 
C_  x  <->  S  C_  (/) ) )
2221elrab 3084 . . . . . 6  |-  ( (/)  e.  { x  e.  J  |  S  C_  x }  <->  (
(/)  e.  J  /\  S  C_  (/) ) )
2322notbii 288 . . . . 5  |-  ( -.  (/)  e.  { x  e.  J  |  S  C_  x }  <->  -.  ( (/)  e.  J  /\  S  C_  (/) ) )
2420, 23bitr2i 242 . . . 4  |-  ( -.  ( (/)  e.  J  /\  S  C_  (/) )  <->  (/)  e/  {
x  e.  J  |  S  C_  x } )
2519, 24sylib 189 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  S  =/=  (/) )  ->  (/)  e/  {
x  e.  J  |  S  C_  x } )
26 sseq2 3362 . . . . . . 7  |-  ( x  =  r  ->  ( S  C_  x  <->  S  C_  r
) )
2726elrab 3084 . . . . . 6  |-  ( r  e.  { x  e.  J  |  S  C_  x }  <->  ( r  e.  J  /\  S  C_  r ) )
28 sseq2 3362 . . . . . . 7  |-  ( x  =  s  ->  ( S  C_  x  <->  S  C_  s
) )
2928elrab 3084 . . . . . 6  |-  ( s  e.  { x  e.  J  |  S  C_  x }  <->  ( s  e.  J  /\  S  C_  s ) )
3027, 29anbi12i 679 . . . . 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 444 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  J  e.  Top )
32 simprll 739 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  r  e.  J )
33 simprrl 741 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  s  e.  J )
34 inopn 16964 . . . . . . . . . . 11  |-  ( ( J  e.  Top  /\  r  e.  J  /\  s  e.  J )  ->  ( r  i^i  s
)  e.  J )
3531, 32, 33, 34syl3anc 1184 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  ( r  i^i  s )  e.  J
)
36 ssin 3555 . . . . . . . . . . . . 13  |-  ( ( S  C_  r  /\  S  C_  s )  <->  S  C_  (
r  i^i  s )
)
3736biimpi 187 . . . . . . . . . . . 12  |-  ( ( S  C_  r  /\  S  C_  s )  ->  S  C_  ( r  i^i  s ) )
3837ad2ant2l 727 . . . . . . . . . . 11  |-  ( ( ( r  e.  J  /\  S  C_  r )  /\  ( s  e.  J  /\  S  C_  s ) )  ->  S  C_  ( r  i^i  s ) )
3938adantl 453 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  ( ( r  e.  J  /\  S  C_  r )  /\  (
s  e.  J  /\  S  C_  s ) ) )  ->  S  C_  (
r  i^i  s )
)
4035, 39jca 519 . . . . . . . . 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 1119 . . . . . . . 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 3362 . . . . . . . . 9  |-  ( x  =  ( r  i^i  s )  ->  ( S  C_  x  <->  S  C_  (
r  i^i  s )
) )
4342elrab 3084 . . . . . . . 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 204 . . . . . . 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 3359 . . . . . . 7  |-  ( r  i^i  s )  C_  ( r  i^i  s
)
46 sseq1 3361 . . . . . . . 8  |-  ( t  =  ( r  i^i  s )  ->  (
t  C_  ( r  i^i  s )  <->  ( r  i^i  s )  C_  (
r  i^i  s )
) )
4746rspcev 3044 . . . . . . 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 644 . . . . . 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 424 . . . . 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 209 . . . 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 2789 . . 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 1134 . 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 17859 . . . 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 978 . 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 889 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    e/ wnel 2599   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   ` cfv 5446   fBascfbas 16681   Topctop 16950
This theorem is referenced by:  neifg  26391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-fbas 16691  df-top 16955
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