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Theorem tailfb 26296
Description: The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypothesis
Ref Expression
tailfb.1  |-  X  =  dom  D
Assertion
Ref Expression
tailfb  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  e.  ( fBas `  X
) )

Proof of Theorem tailfb
Dummy variables  v  u  w  x  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tailfb.1 . . . . 5  |-  X  =  dom  D
21tailf 26294 . . . 4  |-  ( D  e.  DirRel  ->  ( tail `  D
) : X --> ~P X
)
3 frn 5556 . . . 4  |-  ( (
tail `  D ) : X --> ~P X  ->  ran  ( tail `  D
)  C_  ~P X
)
42, 3syl 16 . . 3  |-  ( D  e.  DirRel  ->  ran  ( tail `  D )  C_  ~P X )
54adantr 452 . 2  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  C_ 
~P X )
6 n0 3597 . . . . 5  |-  ( X  =/=  (/)  <->  E. x  x  e.  X )
7 ffn 5550 . . . . . . . 8  |-  ( (
tail `  D ) : X --> ~P X  -> 
( tail `  D )  Fn  X )
8 fnfvelrn 5826 . . . . . . . . 9  |-  ( ( ( tail `  D
)  Fn  X  /\  x  e.  X )  ->  ( ( tail `  D
) `  x )  e.  ran  ( tail `  D
) )
98ex 424 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( x  e.  X  ->  (
( tail `  D ) `  x )  e.  ran  ( tail `  D )
) )
102, 7, 93syl 19 . . . . . . 7  |-  ( D  e.  DirRel  ->  ( x  e.  X  ->  ( ( tail `  D ) `  x )  e.  ran  ( tail `  D )
) )
11 ne0i 3594 . . . . . . 7  |-  ( ( ( tail `  D
) `  x )  e.  ran  ( tail `  D
)  ->  ran  ( tail `  D )  =/=  (/) )
1210, 11syl6 31 . . . . . 6  |-  ( D  e.  DirRel  ->  ( x  e.  X  ->  ran  ( tail `  D )  =/=  (/) ) )
1312exlimdv 1643 . . . . 5  |-  ( D  e.  DirRel  ->  ( E. x  x  e.  X  ->  ran  ( tail `  D
)  =/=  (/) ) )
146, 13syl5bi 209 . . . 4  |-  ( D  e.  DirRel  ->  ( X  =/=  (/)  ->  ran  ( tail `  D )  =/=  (/) ) )
1514imp 419 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  =/=  (/) )
161tailini 26295 . . . . . . . 8  |-  ( ( D  e.  DirRel  /\  x  e.  X )  ->  x  e.  ( ( tail `  D
) `  x )
)
17 n0i 3593 . . . . . . . 8  |-  ( x  e.  ( ( tail `  D ) `  x
)  ->  -.  (
( tail `  D ) `  x )  =  (/) )
1816, 17syl 16 . . . . . . 7  |-  ( ( D  e.  DirRel  /\  x  e.  X )  ->  -.  ( ( tail `  D
) `  x )  =  (/) )
1918nrexdv 2769 . . . . . 6  |-  ( D  e.  DirRel  ->  -.  E. x  e.  X  ( ( tail `  D ) `  x )  =  (/) )
2019adantr 452 . . . . 5  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  -.  E. x  e.  X  ( ( tail `  D
) `  x )  =  (/) )
21 fvelrnb 5733 . . . . . . 7  |-  ( (
tail `  D )  Fn  X  ->  ( (/)  e.  ran  ( tail `  D
)  <->  E. x  e.  X  ( ( tail `  D
) `  x )  =  (/) ) )
222, 7, 213syl 19 . . . . . 6  |-  ( D  e.  DirRel  ->  ( (/)  e.  ran  ( tail `  D )  <->  E. x  e.  X  ( ( tail `  D
) `  x )  =  (/) ) )
2322adantr 452 . . . . 5  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ( (/) 
e.  ran  ( tail `  D )  <->  E. x  e.  X  ( ( tail `  D ) `  x )  =  (/) ) )
2420, 23mtbird 293 . . . 4  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  -.  (/) 
e.  ran  ( tail `  D ) )
25 df-nel 2570 . . . 4  |-  ( (/)  e/ 
ran  ( tail `  D
)  <->  -.  (/)  e.  ran  ( tail `  D )
)
2624, 25sylibr 204 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  (/)  e/  ran  ( tail `  D )
)
27 fvelrnb 5733 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( x  e.  ran  ( tail `  D )  <->  E. u  e.  X  ( ( tail `  D ) `  u )  =  x ) )
28 fvelrnb 5733 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( y  e.  ran  ( tail `  D )  <->  E. v  e.  X  ( ( tail `  D ) `  v )  =  y ) )
2927, 28anbi12d 692 . . . . . . 7  |-  ( (
tail `  D )  Fn  X  ->  ( ( x  e.  ran  ( tail `  D )  /\  y  e.  ran  ( tail `  D ) )  <->  ( E. u  e.  X  (
( tail `  D ) `  u )  =  x  /\  E. v  e.  X  ( ( tail `  D ) `  v
)  =  y ) ) )
302, 7, 293syl 19 . . . . . 6  |-  ( D  e.  DirRel  ->  ( ( x  e.  ran  ( tail `  D )  /\  y  e.  ran  ( tail `  D
) )  <->  ( E. u  e.  X  (
( tail `  D ) `  u )  =  x  /\  E. v  e.  X  ( ( tail `  D ) `  v
)  =  y ) ) )
31 reeanv 2835 . . . . . . 7  |-  ( E. u  e.  X  E. v  e.  X  (
( ( tail `  D
) `  u )  =  x  /\  (
( tail `  D ) `  v )  =  y )  <->  ( E. u  e.  X  ( ( tail `  D ) `  u )  =  x  /\  E. v  e.  X  ( ( tail `  D ) `  v
)  =  y ) )
321dirge 14637 . . . . . . . . . . 11  |-  ( ( D  e.  DirRel  /\  u  e.  X  /\  v  e.  X )  ->  E. w  e.  X  ( u D w  /\  v D w ) )
33323expb 1154 . . . . . . . . . 10  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  E. w  e.  X  ( u D w  /\  v D w ) )
342, 7syl 16 . . . . . . . . . . . . 13  |-  ( D  e.  DirRel  ->  ( tail `  D
)  Fn  X )
35 fnfvelrn 5826 . . . . . . . . . . . . 13  |-  ( ( ( tail `  D
)  Fn  X  /\  w  e.  X )  ->  ( ( tail `  D
) `  w )  e.  ran  ( tail `  D
) )
3634, 35sylan 458 . . . . . . . . . . . 12  |-  ( ( D  e.  DirRel  /\  w  e.  X )  ->  (
( tail `  D ) `  w )  e.  ran  ( tail `  D )
)
3736ad2ant2r 728 . . . . . . . . . . 11  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( ( tail `  D
) `  w )  e.  ran  ( tail `  D
) )
38 vex 2919 . . . . . . . . . . . . . . . . . . . . . 22  |-  x  e. 
_V
39 dirtr 14636 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( D  e.  DirRel  /\  x  e.  _V )  /\  ( u D w  /\  w D x ) )  ->  u D x )
4039exp32 589 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( D  e.  DirRel  /\  x  e.  _V )  ->  (
u D w  -> 
( w D x  ->  u D x ) ) )
4138, 40mpan2 653 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  DirRel  ->  ( u D w  ->  ( w D x  ->  u D x ) ) )
4241com23 74 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  DirRel  ->  ( w D x  ->  ( u D w  ->  u D x ) ) )
4342imp 419 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  DirRel  /\  w D x )  -> 
( u D w  ->  u D x ) )
4443ad2ant2rl 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  w D x ) )  ->  ( u D w  ->  u D x ) )
45 dirtr 14636 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( D  e.  DirRel  /\  x  e.  _V )  /\  ( v D w  /\  w D x ) )  ->  v D x )
4645exp32 589 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( D  e.  DirRel  /\  x  e.  _V )  ->  (
v D w  -> 
( w D x  ->  v D x ) ) )
4738, 46mpan2 653 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  DirRel  ->  ( v D w  ->  ( w D x  ->  v D x ) ) )
4847com23 74 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  DirRel  ->  ( w D x  ->  ( v D w  ->  v D x ) ) )
4948imp 419 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  DirRel  /\  w D x )  -> 
( v D w  ->  v D x ) )
5049ad2ant2rl 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  w D x ) )  ->  ( v D w  ->  v D x ) )
5144, 50anim12d 547 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  w D x ) )  ->  ( ( u D w  /\  v D w )  -> 
( u D x  /\  v D x ) ) )
5251expr 599 . . . . . . . . . . . . . . . 16  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  w  e.  X )  ->  (
w D x  -> 
( ( u D w  /\  v D w )  ->  (
u D x  /\  v D x ) ) ) )
5352com23 74 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  w  e.  X )  ->  (
( u D w  /\  v D w )  ->  ( w D x  ->  ( u D x  /\  v D x ) ) ) )
5453impr 603 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( w D x  ->  ( u D x  /\  v D x ) ) )
551eltail 26293 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  w  e.  X  /\  x  e.  _V )  ->  (
x  e.  ( (
tail `  D ) `  w )  <->  w D x ) )
5638, 55mp3an3 1268 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  DirRel  /\  w  e.  X )  ->  (
x  e.  ( (
tail `  D ) `  w )  <->  w D x ) )
5756ad2ant2r 728 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( x  e.  ( ( tail `  D
) `  w )  <->  w D x ) )
581eltail 26293 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  DirRel  /\  u  e.  X  /\  x  e.  _V )  ->  (
x  e.  ( (
tail `  D ) `  u )  <->  u D x ) )
5938, 58mp3an3 1268 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  DirRel  /\  u  e.  X )  ->  (
x  e.  ( (
tail `  D ) `  u )  <->  u D x ) )
6059adantrr 698 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  ( x  e.  ( ( tail `  D
) `  u )  <->  u D x ) )
611eltail 26293 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  DirRel  /\  v  e.  X  /\  x  e.  _V )  ->  (
x  e.  ( (
tail `  D ) `  v )  <->  v D x ) )
6238, 61mp3an3 1268 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  DirRel  /\  v  e.  X )  ->  (
x  e.  ( (
tail `  D ) `  v )  <->  v D x ) )
6362adantrl 697 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  ( x  e.  ( ( tail `  D
) `  v )  <->  v D x ) )
6460, 63anbi12d 692 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  ( (
x  e.  ( (
tail `  D ) `  u )  /\  x  e.  ( ( tail `  D
) `  v )
)  <->  ( u D x  /\  v D x ) ) )
6564adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( ( x  e.  ( ( tail `  D
) `  u )  /\  x  e.  (
( tail `  D ) `  v ) )  <->  ( u D x  /\  v D x ) ) )
6654, 57, 653imtr4d 260 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( x  e.  ( ( tail `  D
) `  w )  ->  ( x  e.  ( ( tail `  D
) `  u )  /\  x  e.  (
( tail `  D ) `  v ) ) ) )
67 elin 3490 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( (
tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) )  <->  ( x  e.  ( ( tail `  D
) `  u )  /\  x  e.  (
( tail `  D ) `  v ) ) )
6866, 67syl6ibr 219 . . . . . . . . . . . 12  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( x  e.  ( ( tail `  D
) `  w )  ->  x  e.  ( ( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
) ) )
6968ssrdv 3314 . . . . . . . . . . 11  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  -> 
( ( tail `  D
) `  w )  C_  ( ( ( tail `  D ) `  u
)  i^i  ( ( tail `  D ) `  v ) ) )
70 sseq1 3329 . . . . . . . . . . . 12  |-  ( z  =  ( ( tail `  D ) `  w
)  ->  ( z  C_  ( ( ( tail `  D ) `  u
)  i^i  ( ( tail `  D ) `  v ) )  <->  ( ( tail `  D ) `  w )  C_  (
( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
) ) )
7170rspcev 3012 . . . . . . . . . . 11  |-  ( ( ( ( tail `  D
) `  w )  e.  ran  ( tail `  D
)  /\  ( ( tail `  D ) `  w )  C_  (
( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
) )  ->  E. z  e.  ran  ( tail `  D
) z  C_  (
( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
) )
7237, 69, 71syl2anc 643 . . . . . . . . . 10  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  ( u D w  /\  v D w ) ) )  ->  E. z  e.  ran  ( tail `  D )
z  C_  ( (
( tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) ) )
7333, 72rexlimddv 2794 . . . . . . . . 9  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  E. z  e.  ran  ( tail `  D
) z  C_  (
( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
) )
74 ineq1 3495 . . . . . . . . . . . 12  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( ( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
)  =  ( x  i^i  ( ( tail `  D ) `  v
) ) )
7574sseq2d 3336 . . . . . . . . . . 11  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( z 
C_  ( ( (
tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) )  <->  z  C_  ( x  i^i  (
( tail `  D ) `  v ) ) ) )
7675rexbidv 2687 . . . . . . . . . 10  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( E. z  e.  ran  ( tail `  D ) z 
C_  ( ( (
tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) )  <->  E. z  e.  ran  ( tail `  D
) z  C_  (
x  i^i  ( ( tail `  D ) `  v ) ) ) )
77 ineq2 3496 . . . . . . . . . . . 12  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( x  i^i  ( ( tail `  D ) `  v
) )  =  ( x  i^i  y ) )
7877sseq2d 3336 . . . . . . . . . . 11  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( z 
C_  ( x  i^i  ( ( tail `  D
) `  v )
)  <->  z  C_  (
x  i^i  y )
) )
7978rexbidv 2687 . . . . . . . . . 10  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( E. z  e.  ran  ( tail `  D ) z 
C_  ( x  i^i  ( ( tail `  D
) `  v )
)  <->  E. z  e.  ran  ( tail `  D )
z  C_  ( x  i^i  y ) ) )
8076, 79sylan9bb 681 . . . . . . . . 9  |-  ( ( ( ( tail `  D
) `  u )  =  x  /\  (
( tail `  D ) `  v )  =  y )  ->  ( E. z  e.  ran  ( tail `  D ) z  C_  ( ( ( tail `  D ) `  u
)  i^i  ( ( tail `  D ) `  v ) )  <->  E. z  e.  ran  ( tail `  D
) z  C_  (
x  i^i  y )
) )
8173, 80syl5ibcom 212 . . . . . . . 8  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  ( (
( ( tail `  D
) `  u )  =  x  /\  (
( tail `  D ) `  v )  =  y )  ->  E. z  e.  ran  ( tail `  D
) z  C_  (
x  i^i  y )
) )
8281rexlimdvva 2797 . . . . . . 7  |-  ( D  e.  DirRel  ->  ( E. u  e.  X  E. v  e.  X  ( (
( tail `  D ) `  u )  =  x  /\  ( ( tail `  D ) `  v
)  =  y )  ->  E. z  e.  ran  ( tail `  D )
z  C_  ( x  i^i  y ) ) )
8331, 82syl5bir 210 . . . . . 6  |-  ( D  e.  DirRel  ->  ( ( E. u  e.  X  ( ( tail `  D
) `  u )  =  x  /\  E. v  e.  X  ( ( tail `  D ) `  v )  =  y )  ->  E. z  e.  ran  ( tail `  D
) z  C_  (
x  i^i  y )
) )
8430, 83sylbid 207 . . . . 5  |-  ( D  e.  DirRel  ->  ( ( x  e.  ran  ( tail `  D )  /\  y  e.  ran  ( tail `  D
) )  ->  E. z  e.  ran  ( tail `  D
) z  C_  (
x  i^i  y )
) )
8584adantr 452 . . . 4  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  (
( x  e.  ran  ( tail `  D )  /\  y  e.  ran  ( tail `  D )
)  ->  E. z  e.  ran  ( tail `  D
) z  C_  (
x  i^i  y )
) )
8685ralrimivv 2757 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  A. x  e.  ran  ( tail `  D
) A. y  e. 
ran  ( tail `  D
) E. z  e. 
ran  ( tail `  D
) z  C_  (
x  i^i  y )
)
8715, 26, 863jca 1134 . 2  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ( ran  ( tail `  D
)  =/=  (/)  /\  (/)  e/  ran  ( tail `  D )  /\  A. x  e.  ran  ( tail `  D ) A. y  e.  ran  ( tail `  D ) E. z  e.  ran  ( tail `  D )
z  C_  ( x  i^i  y ) ) )
88 dmexg 5089 . . . . 5  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
891, 88syl5eqel 2488 . . . 4  |-  ( D  e.  DirRel  ->  X  e.  _V )
9089adantr 452 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  X  e.  _V )
91 isfbas2 17820 . . 3  |-  ( X  e.  _V  ->  ( ran  ( tail `  D
)  e.  ( fBas `  X )  <->  ( ran  ( tail `  D )  C_ 
~P X  /\  ( ran  ( tail `  D
)  =/=  (/)  /\  (/)  e/  ran  ( tail `  D )  /\  A. x  e.  ran  ( tail `  D ) A. y  e.  ran  ( tail `  D ) E. z  e.  ran  ( tail `  D )
z  C_  ( x  i^i  y ) ) ) ) )
9290, 91syl 16 . 2  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ( ran  ( tail `  D
)  e.  ( fBas `  X )  <->  ( ran  ( tail `  D )  C_ 
~P X  /\  ( ran  ( tail `  D
)  =/=  (/)  /\  (/)  e/  ran  ( tail `  D )  /\  A. x  e.  ran  ( tail `  D ) A. y  e.  ran  ( tail `  D ) E. z  e.  ran  ( tail `  D )
z  C_  ( x  i^i  y ) ) ) ) )
935, 87, 92mpbir2and 889 1  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  e.  ( fBas `  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567    e/ wnel 2568   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   class class class wbr 4172   dom cdm 4837   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413   DirRelcdir 14628   tailctail 14629   fBascfbas 16644
This theorem is referenced by:  filnetlem4  26300
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-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-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  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-dir 14630  df-tail 14631  df-fbas 16654
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