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Theorem neipcfilu 20534
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 997 . . . . 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 19204 . . . . 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 998 . . . . 5  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  P  e.  X )
76snssd 4172 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  C_  X )
8 snnzg 4144 . . . . 5  |-  ( P  e.  X  ->  { P }  =/=  (/) )
96, 8syl 16 . . . 4  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  { P }  =/=  (/) )
10 neifil 20116 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { P }  C_  X  /\  { P }  =/=  (/) )  -> 
( ( nei `  J
) `  { P } )  e.  ( Fil `  X ) )
115, 7, 9, 10syl3anc 1228 . . 3  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  ( ( nei `  J ) `  { P } )  e.  ( Fil `  X
) )
12 filfbas 20084 . . 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 2467 . . . . . . . . . 10  |-  ( w
" { P }
)  =  ( w
" { P }
)
15 imaeq1 5330 . . . . . . . . . . . 12  |-  ( v  =  w  ->  (
v " { P } )  =  ( w " { P } ) )
1615eqeq2d 2481 . . . . . . . . . . 11  |-  ( v  =  w  ->  (
( w " { P } )  =  ( v " { P } )  <->  ( w " { P } )  =  ( w " { P } ) ) )
1716rspcev 3214 . . . . . . . . . 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 3116 . . . . . . . . . 10  |-  w  e. 
_V
20 imaexg 6718 . . . . . . . . . 10  |-  ( w  e.  _V  ->  (
w " { P } )  e.  _V )
21 eqid 2467 . . . . . . . . . . 11  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
2221elrnmpt 5247 . . . . . . . . . 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 20499 . . . . . . . . . . . 12  |-  ( W  e. UnifSp 
<->  ( U  e.  (UnifOn `  X )  /\  J  =  (unifTop `  U )
) )
2827simplbi 460 . . . . . . . . . . 11  |-  ( W  e. UnifSp  ->  U  e.  (UnifOn `  X ) )
29283ad2ant1 1017 . . . . . . . . . 10  |-  ( ( W  e. UnifSp  /\  W  e. 
TopSp  /\  P  e.  X
)  ->  U  e.  (UnifOn `  X ) )
30 eqid 2467 . . . . . . . . . . 11  |-  (unifTop `  U
)  =  (unifTop `  U
)
3130utopsnneip 20486 . . . . . . . . . 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 2537 . . . . . . . 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 999 . . . . . . . . . 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 1218 . . . . . . . . 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 5868 . . . . . . 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 5866 . . . . . 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 2558 . . . . 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 5024 . . . . . . 7  |-  ( a  =  ( w " { P } )  -> 
( a  X.  a
)  =  ( ( w " { P } )  X.  (
w " { P } ) ) )
4645sseq1d 3531 . . . . . 6  |-  ( a  =  ( w " { P } )  -> 
( ( a  X.  a )  C_  v  <->  ( ( w " { P } )  X.  (
w " { P } ) )  C_  v ) )
4746rspcev 3214 . . . . 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 1035 . . . . . . . 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 20453 . . . . . . . 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 20442 . . . . . . . . . . . 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 1036 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 20461 . . . . . . . . . . 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 5163 . . . . . . . . . . 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 5156 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
w  o.  w ) 
C_  ( u  o.  w ) )
67 coss2 5157 . . . . . . . . . . . . . 14  |-  ( w 
C_  u  ->  (
u  o.  w ) 
C_  ( u  o.  u ) )
6866, 67sstrd 3514 . . . . . . . . . . . . 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 3514 . . . . . . . . . . 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 3538 . . . . . . . . . 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 3514 . . . . . . . . 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 2938 . . . . . . 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 20448 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
78773adant2 1015 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X  /\  v  e.  U )  ->  E. u  e.  U  ( u  o.  u )  C_  v
)
7976, 78r19.29a 3003 . . . . 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 1228 . . . 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 3003 . . 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 2878 . 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 20526 . . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   ran crn 5000   "cima 5002    o. ccom 5003   ` cfv 5586   Basecbs 14486   TopOpenctopn 14673   fBascfbas 18177  TopOnctopon 19162   TopSpctps 19164   neicnei 19364   Filcfil 20081  UnifOncust 20437  unifTopcutop 20468  UnifStcuss 20491  UnifSpcusp 20492  CauFiluccfilu 20524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-fbas 18187  df-top 19166  df-topon 19169  df-topsp 19170  df-nei 19365  df-fil 20082  df-ust 20438  df-utop 20469  df-usp 20495  df-cfilu 20525
This theorem is referenced by:  ucnextcn  20542
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