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Theorem neipcfilu 18279
Description: In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)
Hypotheses
Ref Expression
neipcfilu.x  |-  X  =  ( Base `  W
)
neipcfilu.j  |-  J  =  ( TopOpen `  W )
neipcfilu.u  |-  U  =  (UnifSt `  W )
Assertion
Ref Expression
neipcfilu  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  (CauFilu `  U ) )

Proof of Theorem neipcfilu
Dummy variables  v 
a  w  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  W  e.  TopSp
)
2 neipcfilu.x . . . . . 6  |-  X  =  ( Base `  W
)
3 neipcfilu.j . . . . . 6  |-  J  =  ( TopOpen `  W )
42, 3istps 16956 . . . . 5  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
51, 4sylib 189 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  J  e.  (TopOn `  X ) )
6 simp3 959 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  P  e.  X )
76snssd 3903 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  C_  X )
8 snnzg 3881 . . . . 5  |-  ( P  e.  X  ->  { P }  =/=  (/) )
96, 8syl 16 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  =/=  (/) )
10 neifil 17865 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { P }  C_  X  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  X ) )
115, 7, 9, 10syl3anc 1184 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  ( Fil `  X
) )
12 filfbas 17833 . . 3  |-  ( ( ( nei `  J
) `  { P } )  e.  ( Fil `  X )  ->  ( ( nei `  J ) `  { P } )  e.  (
fBas `  X )
)
1311, 12syl 16 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  ( fBas `  X
) )
14 eqid 2404 . . . . . . . . . 10  |-  ( w
" { P }
)  =  ( w
" { P }
)
15 imaeq1 5157 . . . . . . . . . . . 12  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
1615eqeq2d 2415 . . . . . . . . . . 11  |-  ( v  =  w  ->  (
( w " { P } )  =  ( v " { P } )  <->  ( w " { P } )  =  ( w " { P } ) ) )
1716rspcev 3012 . . . . . . . . . 10  |-  ( ( w  e.  U  /\  ( w " { P } )  =  ( w " { P } ) )  ->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
1814, 17mpan2 653 . . . . . . . . 9  |-  ( w  e.  U  ->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
19 vex 2919 . . . . . . . . . 10  |-  w  e. 
_V
20 imaexg 5176 . . . . . . . . . 10  |-  ( w  e.  _V  ->  (
w " { P } )  e.  _V )
21 eqid 2404 . . . . . . . . . . 11  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
2221elrnmpt 5076 . . . . . . . . . 10  |-  ( ( w " { P } )  e.  _V  ->  ( ( w " { P } )  e. 
ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) ) )
2319, 20, 22mp2b 10 . . . . . . . . 9  |-  ( ( w " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
2418, 23sylibr 204 . . . . . . . 8  |-  ( w  e.  U  ->  (
w " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2524ad2antlr 708 . . . . . . 7  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  (
w " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) )
26 neipcfilu.u . . . . . . . . . . . . 13  |-  U  =  (UnifSt `  W )
272, 26, 3isusp 18244 . . . . . . . . . . . 12  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  X )  /\  J  =  (unifTop `  U )
) )
2827simplbi 447 . . . . . . . . . . 11  |-  ( W  e. UnifSp  ->  U  e.  (UnifOn `  X ) )
29283ad2ant1 978 . . . . . . . . . 10  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  U  e.  (UnifOn `  X ) )
30 eqid 2404 . . . . . . . . . . 11  |-  (unifTop `  U
)  =  (unifTop `  U
)
3130utopsnneip 18231 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  (unifTop `  U ) ) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
3229, 6, 31syl2anc 643 . . . . . . . . 9  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  (unifTop `  U
) ) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
3332eleq2d 2471 . . . . . . . 8  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( (
w " { P } )  e.  ( ( nei `  (unifTop `  U ) ) `  { P } )  <->  ( w " { P } )  e.  ran  ( v  e.  U  |->  ( v
" { P }
) ) ) )
3433ad3antrrr 711 . . . . . . 7  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  (
( w " { P } )  e.  ( ( nei `  (unifTop `  U ) ) `  { P } )  <->  ( w " { P } )  e.  ran  ( v  e.  U  |->  ( v
" { P }
) ) ) )
3525, 34mpbird 224 . . . . . 6  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  (
w " { P } )  e.  ( ( nei `  (unifTop `  U ) ) `  { P } ) )
36 simpl1 960 . . . . . . . . . 10  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  (
v  e.  U  /\  w  e.  U  /\  ( ( w " { P } )  X.  ( w " { P } ) )  C_  v ) )  ->  W  e. UnifSp )
37363anassrs 1175 . . . . . . . . 9  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  W  e. UnifSp )
3827simprbi 451 . . . . . . . . 9  |-  ( W  e. UnifSp  ->  J  =  (unifTop `  U ) )
3937, 38syl 16 . . . . . . . 8  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  J  =  (unifTop `  U )
)
4039fveq2d 5691 . . . . . . 7  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  ( nei `  J )  =  ( nei `  (unifTop `  U ) ) )
4140fveq1d 5689 . . . . . 6  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  (
( nei `  J
) `  { P } )  =  ( ( nei `  (unifTop `  U ) ) `  { P } ) )
4235, 41eleqtrrd 2481 . . . . 5  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  (
w " { P } )  e.  ( ( nei `  J
) `  { P } ) )
43 simpr 448 . . . . 5  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )
44 id 20 . . . . . . . 8  |-  ( a  =  ( w " { P } )  -> 
a  =  ( w
" { P }
) )
4544, 44xpeq12d 4862 . . . . . . 7  |-  ( a  =  ( w " { P } )  -> 
( a  X.  a
)  =  ( ( w " { P } )  X.  (
w " { P } ) ) )
4645sseq1d 3335 . . . . . 6  |-  ( a  =  ( w " { P } )  -> 
( ( a  X.  a )  C_  v  <->  ( ( w " { P } )  X.  (
w " { P } ) )  C_  v ) )
4746rspcev 3012 . . . . 5  |-  ( ( ( w " { P } )  e.  ( ( nei `  J
) `  { P } )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  E. a  e.  ( ( nei `  J
) `  { P } ) ( a  X.  a )  C_  v )
4842, 43, 47syl2anc 643 . . . 4  |-  ( ( ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U
)  /\  w  e.  U )  /\  (
( w " { P } )  X.  (
w " { P } ) )  C_  v )  ->  E. a  e.  ( ( nei `  J
) `  { P } ) ( a  X.  a )  C_  v )
4929adantr 452 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  U  e.  (UnifOn `  X )
)
506adantr 452 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  P  e.  X )
51 simpr 448 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  v  e.  U )
52 simpll1 996 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  ->  U  e.  (UnifOn `  X
) )
53 simplr 732 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  ->  u  e.  U )
54 ustexsym 18198 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  u  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  u ) )
5552, 53, 54syl2anc 643 . . . . . . 7  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  u ) )
5652ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  ->  U  e.  (UnifOn `  X
) )
57 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  ->  w  e.  U )
58 ustssxp 18187 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( X  X.  X
) )
5956, 57, 58syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  ->  w  C_  ( X  X.  X ) )
60 simpll2 997 . . . . . . . . . . . 12  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
( u  o.  u
)  C_  v  /\  w  e.  U  /\  ( `' w  =  w  /\  w  C_  u ) ) )  ->  P  e.  X )
61603anassrs 1175 . . . . . . . . . . 11  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  ->  P  e.  X )
62 ustneism 18206 . . . . . . . . . . 11  |-  ( ( w  C_  ( X  X.  X )  /\  P  e.  X )  ->  (
( w " { P } )  X.  (
w " { P } ) )  C_  ( w  o.  `' w ) )
6359, 61, 62syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( ( w " { P } )  X.  ( w " { P } ) )  C_  ( w  o.  `' w ) )
64 simprl 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  ->  `' w  =  w
)
6564coeq2d 4994 . . . . . . . . . . 11  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( w  o.  `' w )  =  ( w  o.  w ) )
66 coss1 4987 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  w ) )
67 coss2 4988 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
u  o.  w ) 
C_  ( u  o.  u ) )
6866, 67sstrd 3318 . . . . . . . . . . . . 13  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  u ) )
6968ad2antll 710 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( w  o.  w
)  C_  ( u  o.  u ) )
70 simpllr 736 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( u  o.  u
)  C_  v )
7169, 70sstrd 3318 . . . . . . . . . . 11  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( w  o.  w
)  C_  v )
7265, 71eqsstrd 3342 . . . . . . . . . 10  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( w  o.  `' w )  C_  v
)
7363, 72sstrd 3318 . . . . . . . . 9  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  u ) )  -> 
( ( w " { P } )  X.  ( w " { P } ) )  C_  v )
7473ex 424 . . . . . . . 8  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  /\  w  e.  U )  ->  ( ( `' w  =  w  /\  w  C_  u )  ->  (
( w " { P } )  X.  (
w " { P } ) )  C_  v ) )
7574reximdva 2778 . . . . . . 7  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  -> 
( E. w  e.  U  ( `' w  =  w  /\  w  C_  u )  ->  E. w  e.  U  ( (
w " { P } )  X.  (
w " { P } ) )  C_  v ) )
7655, 75mpd 15 . . . . . 6  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  ->  E. w  e.  U  ( ( w " { P } )  X.  ( w " { P } ) )  C_  v )
77 ustexhalf 18193 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
78773adant2 976 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
7976, 78r19.29a 2810 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  ->  E. w  e.  U  ( (
w " { P } )  X.  (
w " { P } ) )  C_  v )
8049, 50, 51, 79syl3anc 1184 . . . 4  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  E. w  e.  U  ( (
w " { P } )  X.  (
w " { P } ) )  C_  v )
8148, 80r19.29a 2810 . . 3  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  E. a  e.  ( ( nei `  J
) `  { P } ) ( a  X.  a )  C_  v )
8281ralrimiva 2749 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  A. v  e.  U  E. a  e.  ( ( nei `  J
) `  { P } ) ( a  X.  a )  C_  v )
83 iscfilu 18271 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( (
( nei `  J
) `  { P } )  e.  (CauFilu `  U )  <->  ( (
( nei `  J
) `  { P } )  e.  (
fBas `  X )  /\  A. v  e.  U  E. a  e.  (
( nei `  J
) `  { P } ) ( a  X.  a )  C_  v ) ) )
8429, 83syl 16 . 2  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( (
( nei `  J
) `  { P } )  e.  (CauFilu `  U )  <->  ( (
( nei `  J
) `  { P } )  e.  (
fBas `  X )  /\  A. v  e.  U  E. a  e.  (
( nei `  J
) `  { P } ) ( a  X.  a )  C_  v ) ) )
8513, 82, 84mpbir2and 889 1  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  (CauFilu `  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   (/)c0 3588   {csn 3774    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   ran crn 4838   "cima 4840    o. ccom 4841   ` cfv 5413   Basecbs 13424   TopOpenctopn 13604   fBascfbas 16644  TopOnctopon 16914   TopSpctps 16916   neicnei 17116   Filcfil 17830  UnifOncust 18182  unifTopcutop 18213  UnifStcuss 18236  UnifSpcusp 18237  CauFiluccfilu 18269
This theorem is referenced by:  ucnextcn  18287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-fin 7072  df-fi 7374  df-fbas 16654  df-top 16918  df-topon 16921  df-topsp 16922  df-nei 17117  df-fil 17831  df-ust 18183  df-utop 18214  df-usp 18240  df-cfilu 18270
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