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Theorem neipcfilu 19876
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 989 . . . . 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 18546 . . . . 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 990 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  P  e.  X )
76snssd 4023 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  C_  X )
8 snnzg 3997 . . . . 5  |-  ( P  e.  X  ->  { P }  =/=  (/) )
96, 8syl 16 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  =/=  (/) )
10 neifil 19458 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { P }  C_  X  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  X ) )
115, 7, 9, 10syl3anc 1218 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  ( Fil `  X
) )
12 filfbas 19426 . . 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 2443 . . . . . . . . . 10  |-  ( w
" { P }
)  =  ( w
" { P }
)
15 imaeq1 5169 . . . . . . . . . . . 12  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
1615eqeq2d 2454 . . . . . . . . . . 11  |-  ( v  =  w  ->  (
( w " { P } )  =  ( v " { P } )  <->  ( w " { P } )  =  ( w " { P } ) ) )
1716rspcev 3078 . . . . . . . . . 10  |-  ( ( w  e.  U  /\  ( w " { P } )  =  ( w " { P } ) )  ->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
1814, 17mpan2 671 . . . . . . . . 9  |-  ( w  e.  U  ->  E. v  e.  U  ( w " { P } )  =  ( v " { P } ) )
19 vex 2980 . . . . . . . . . 10  |-  w  e. 
_V
20 imaexg 6520 . . . . . . . . . 10  |-  ( w  e.  _V  ->  (
w " { P } )  e.  _V )
21 eqid 2443 . . . . . . . . . . 11  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
2221elrnmpt 5091 . . . . . . . . . 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 726 . . . . . . 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 19841 . . . . . . . . . . . 12  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  X )  /\  J  =  (unifTop `  U )
) )
2827simplbi 460 . . . . . . . . . . 11  |-  ( W  e. UnifSp  ->  U  e.  (UnifOn `  X ) )
29283ad2ant1 1009 . . . . . . . . . 10  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  U  e.  (UnifOn `  X ) )
30 eqid 2443 . . . . . . . . . . 11  |-  (unifTop `  U
)  =  (unifTop `  U
)
3130utopsnneip 19828 . . . . . . . . . 10  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  (unifTop `  U ) ) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
3229, 6, 31syl2anc 661 . . . . . . . . 9  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  (unifTop `  U
) ) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
3332eleq2d 2510 . . . . . . . 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 729 . . . . . . 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 991 . . . . . . . . . 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 1209 . . . . . . . . 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 464 . . . . . . . . 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 5700 . . . . . . 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 5698 . . . . . 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 2520 . . . . 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 461 . . . . 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 }
) )
4544, 44xpeq12d 4870 . . . . . . 7  |-  ( a  =  ( w " { P } )  -> 
( a  X.  a
)  =  ( ( w " { P } )  X.  (
w " { P } ) ) )
4645sseq1d 3388 . . . . . 6  |-  ( a  =  ( w " { P } )  -> 
( ( a  X.  a )  C_  v  <->  ( ( w " { P } )  X.  (
w " { P } ) )  C_  v ) )
4746rspcev 3078 . . . . 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 661 . . . 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 465 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  U  e.  (UnifOn `  X )
)
506adantr 465 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  P  e.  X )
51 simpr 461 . . . . 5  |-  ( ( ( W  e. UnifSp  /\  W  e.  TopSp  /\  P  e.  X )  /\  v  e.  U )  ->  v  e.  U )
52 simpll1 1027 . . . . . . . 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 754 . . . . . . . 8  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U
)  /\  u  e.  U )  /\  (
u  o.  u ) 
C_  v )  ->  u  e.  U )
54 ustexsym 19795 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  u  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  u ) )
5552, 53, 54syl2anc 661 . . . . . . 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 725 . . . . . . . . . . . 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 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  e.  U )
58 ustssxp 19784 . . . . . . . . . . . 12  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( X  X.  X
) )
5956, 57, 58syl2anc 661 . . . . . . . . . . 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 1028 . . . . . . . . . . . 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 1209 . . . . . . . . . . 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 19803 . . . . . . . . . . 11  |-  ( ( w  C_  ( X  X.  X )  /\  P  e.  X )  ->  (
( w " { P } )  X.  (
w " { P } ) )  C_  ( w  o.  `' w ) )
6359, 61, 62syl2anc 661 . . . . . . . . . 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 755 . . . . . . . . . . . 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 5007 . . . . . . . . . . 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 5000 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  w ) )
67 coss2 5001 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
u  o.  w ) 
C_  ( u  o.  u ) )
6866, 67sstrd 3371 . . . . . . . . . . . . 13  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  u ) )
6968ad2antll 728 . . . . . . . . . . . 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 3371 . . . . . . . . . . 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 3395 . . . . . . . . . 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 3371 . . . . . . . . 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 434 . . . . . . . 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 2833 . . . . . . 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 19790 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
78773adant2 1007 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
7976, 78r19.29a 2867 . . . . 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 1218 . . . 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 2867 . . 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 2804 . 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 19868 . . 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 913 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   {csn 3882    e. cmpt 4355    X. cxp 4843   `'ccnv 4844   ran crn 4846   "cima 4848    o. ccom 4849   ` cfv 5423   Basecbs 14179   TopOpenctopn 14365   fBascfbas 17809  TopOnctopon 18504   TopSpctps 18506   neicnei 18706   Filcfil 19423  UnifOncust 19779  unifTopcutop 19810  UnifStcuss 19833  UnifSpcusp 19834  CauFiluccfilu 19866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-fin 7319  df-fi 7666  df-fbas 17819  df-top 18508  df-topon 18511  df-topsp 18512  df-nei 18707  df-fil 19424  df-ust 19780  df-utop 19811  df-usp 19837  df-cfilu 19867
This theorem is referenced by:  ucnextcn  19884
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