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Theorem neipcfilu 20965
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 995 . . . . 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 19604 . . . . 5  |-  ( W  e.  TopSp 
<->  J  e.  (TopOn `  X ) )
51, 4sylib 196 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  J  e.  (TopOn `  X ) )
6 simp3 996 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  P  e.  X )
76snssd 4161 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  C_  X )
8 snnzg 4133 . . . . 5  |-  ( P  e.  X  ->  { P }  =/=  (/) )
96, 8syl 16 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  =/=  (/) )
10 neifil 20547 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { P }  C_  X  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  X ) )
115, 7, 9, 10syl3anc 1226 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  ( Fil `  X
) )
12 filfbas 20515 . . 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 2454 . . . . . . . . . 10  |-  ( w
" { P }
)  =  ( w
" { P }
)
15 imaeq1 5320 . . . . . . . . . . . 12  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
1615eqeq2d 2468 . . . . . . . . . . 11  |-  ( v  =  w  ->  (
( w " { P } )  =  ( v " { P } )  <->  ( w " { P } )  =  ( w " { P } ) ) )
1716rspcev 3207 . . . . . . . . . 10  |-  ( ( w  e.  U  /\  ( w " { P } )  =  ( w " { P } ) )  ->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
1814, 17mpan2 669 . . . . . . . . 9  |-  ( w  e.  U  ->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
19 vex 3109 . . . . . . . . . 10  |-  w  e. 
_V
20 imaexg 6710 . . . . . . . . . 10  |-  ( w  e.  _V  ->  (
w " { P } )  e.  _V )
21 eqid 2454 . . . . . . . . . . 11  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
2221elrnmpt 5238 . . . . . . . . . 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 212 . . . . . . . 8  |-  ( w  e.  U  ->  (
w " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) ) )
2524ad2antlr 724 . . . . . . 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 20930 . . . . . . . . . . . 12  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  X )  /\  J  =  (unifTop `  U )
) )
2827simplbi 458 . . . . . . . . . . 11  |-  ( W  e. UnifSp  ->  U  e.  (UnifOn `  X ) )
29283ad2ant1 1015 . . . . . . . . . 10  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  U  e.  (UnifOn `  X ) )
30 eqid 2454 . . . . . . . . . . 11  |-  (unifTop `  U
)  =  (unifTop `  U
)
3130utopsnneip 20917 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  (unifTop `  U ) ) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
3229, 6, 31syl2anc 659 . . . . . . . . 9  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  (unifTop `  U
) ) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
3332eleq2d 2524 . . . . . . . 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 727 . . . . . . 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 232 . . . . . 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 997 . . . . . . . . . 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 1216 . . . . . . . . 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 462 . . . . . . . . 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 5852 . . . . . . 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 5850 . . . . . 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 2545 . . . . 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 459 . . . . 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 22 . . . . . . . 8  |-  ( a  =  ( w " { P } )  -> 
a  =  ( w
" { P }
) )
4544sqxpeqd 5014 . . . . . . 7  |-  ( a  =  ( w " { P } )  -> 
( a  X.  a
)  =  ( ( w " { P } )  X.  (
w " { P } ) ) )
4645sseq1d 3516 . . . . . 6  |-  ( a  =  ( w " { P } )  -> 
( ( a  X.  a )  C_  v  <->  ( ( w " { P } )  X.  (
w " { P } ) )  C_  v ) )
4746rspcev 3207 . . . . 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 659 . . . 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 463 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  U  e.  (UnifOn `  X )
)
506adantr 463 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  P  e.  X )
51 simpr 459 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  v  e.  U )
52 simpll1 1033 . . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  ->  u  e.  U )
54 ustexsym 20884 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  u  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  u ) )
5552, 53, 54syl2anc 659 . . . . . . 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 723 . . . . . . . . . . . 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 753 . . . . . . . . . . . 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 20873 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( X  X.  X
) )
5956, 57, 58syl2anc 659 . . . . . . . . . . 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 1034 . . . . . . . . . . . 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 1216 . . . . . . . . . . 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 20892 . . . . . . . . . . 11  |-  ( ( w  C_  ( X  X.  X )  /\  P  e.  X )  ->  (
( w " { P } )  X.  (
w " { P } ) )  C_  ( w  o.  `' w ) )
6359, 61, 62syl2anc 659 . . . . . . . . . 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 754 . . . . . . . . . . . 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 5154 . . . . . . . . . . 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 5147 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  w ) )
67 coss2 5148 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
u  o.  w ) 
C_  ( u  o.  u ) )
6866, 67sstrd 3499 . . . . . . . . . . . . 13  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  u ) )
6968ad2antll 726 . . . . . . . . . . . 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 758 . . . . . . . . . . . 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 3499 . . . . . . . . . . 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 3523 . . . . . . . . . 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 3499 . . . . . . . . 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 432 . . . . . . . 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 2929 . . . . . . 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 20879 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
78773adant2 1013 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
7976, 78r19.29a 2996 . . . . 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 1226 . . . 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 2996 . . 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 2868 . 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 20957 . . 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 920 1  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  (CauFilu `  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    C_ wss 3461   (/)c0 3783   {csn 4016    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   ran crn 4989   "cima 4991    o. ccom 4992   ` cfv 5570   Basecbs 14716   TopOpenctopn 14911   fBascfbas 18601  TopOnctopon 19562   TopSpctps 19564   neicnei 19765   Filcfil 20512  UnifOncust 20868  unifTopcutop 20899  UnifStcuss 20922  UnifSpcusp 20923  CauFiluccfilu 20955
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-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  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-reu 2811  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-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  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-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-fin 7513  df-fi 7863  df-fbas 18611  df-top 19566  df-topon 19569  df-topsp 19570  df-nei 19766  df-fil 20513  df-ust 20869  df-utop 20900  df-usp 20926  df-cfilu 20956
This theorem is referenced by:  ucnextcn  20973
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