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Theorem tailfb 30379
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 30377 . . . 4  |-  ( D  e.  DirRel  ->  ( tail `  D
) : X --> ~P X
)
3 frn 5743 . . . 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 465 . 2  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  C_ 
~P X )
6 n0 3803 . . . . 5  |-  ( X  =/=  (/)  <->  E. x  x  e.  X )
7 ffn 5737 . . . . . . . 8  |-  ( (
tail `  D ) : X --> ~P X  -> 
( tail `  D )  Fn  X )
8 fnfvelrn 6029 . . . . . . . . 9  |-  ( ( ( tail `  D
)  Fn  X  /\  x  e.  X )  ->  ( ( tail `  D
) `  x )  e.  ran  ( tail `  D
) )
98ex 434 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( x  e.  X  ->  (
( tail `  D ) `  x )  e.  ran  ( tail `  D )
) )
102, 7, 93syl 20 . . . . . . 7  |-  ( D  e.  DirRel  ->  ( x  e.  X  ->  ( ( tail `  D ) `  x )  e.  ran  ( tail `  D )
) )
11 ne0i 3799 . . . . . . 7  |-  ( ( ( tail `  D
) `  x )  e.  ran  ( tail `  D
)  ->  ran  ( tail `  D )  =/=  (/) )
1210, 11syl6 33 . . . . . 6  |-  ( D  e.  DirRel  ->  ( x  e.  X  ->  ran  ( tail `  D )  =/=  (/) ) )
1312exlimdv 1725 . . . . 5  |-  ( D  e.  DirRel  ->  ( E. x  x  e.  X  ->  ran  ( tail `  D
)  =/=  (/) ) )
146, 13syl5bi 217 . . . 4  |-  ( D  e.  DirRel  ->  ( X  =/=  (/)  ->  ran  ( tail `  D )  =/=  (/) ) )
1514imp 429 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  =/=  (/) )
161tailini 30378 . . . . . . . 8  |-  ( ( D  e.  DirRel  /\  x  e.  X )  ->  x  e.  ( ( tail `  D
) `  x )
)
17 n0i 3798 . . . . . . . 8  |-  ( x  e.  ( ( tail `  D ) `  x
)  ->  -.  (
( tail `  D ) `  x )  =  (/) )
1816, 17syl 16 . . . . . . 7  |-  ( ( D  e.  DirRel  /\  x  e.  X )  ->  -.  ( ( tail `  D
) `  x )  =  (/) )
1918nrexdv 2913 . . . . . 6  |-  ( D  e.  DirRel  ->  -.  E. x  e.  X  ( ( tail `  D ) `  x )  =  (/) )
2019adantr 465 . . . . 5  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  -.  E. x  e.  X  ( ( tail `  D
) `  x )  =  (/) )
21 fvelrnb 5920 . . . . . . 7  |-  ( (
tail `  D )  Fn  X  ->  ( (/)  e.  ran  ( tail `  D
)  <->  E. x  e.  X  ( ( tail `  D
) `  x )  =  (/) ) )
222, 7, 213syl 20 . . . . . 6  |-  ( D  e.  DirRel  ->  ( (/)  e.  ran  ( tail `  D )  <->  E. x  e.  X  ( ( tail `  D
) `  x )  =  (/) ) )
2322adantr 465 . . . . 5  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ( (/) 
e.  ran  ( tail `  D )  <->  E. x  e.  X  ( ( tail `  D ) `  x )  =  (/) ) )
2420, 23mtbird 301 . . . 4  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  -.  (/) 
e.  ran  ( tail `  D ) )
25 df-nel 2655 . . . 4  |-  ( (/)  e/ 
ran  ( tail `  D
)  <->  -.  (/)  e.  ran  ( tail `  D )
)
2624, 25sylibr 212 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  (/)  e/  ran  ( tail `  D )
)
27 fvelrnb 5920 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( x  e.  ran  ( tail `  D )  <->  E. u  e.  X  ( ( tail `  D ) `  u )  =  x ) )
28 fvelrnb 5920 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( y  e.  ran  ( tail `  D )  <->  E. v  e.  X  ( ( tail `  D ) `  v )  =  y ) )
2927, 28anbi12d 710 . . . . . . 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 20 . . . . . 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 3025 . . . . . . 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 15994 . . . . . . . . . . 11  |-  ( ( D  e.  DirRel  /\  u  e.  X  /\  v  e.  X )  ->  E. w  e.  X  ( u D w  /\  v D w ) )
33323expb 1197 . . . . . . . . . 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 6029 . . . . . . . . . . . . 13  |-  ( ( ( tail `  D
)  Fn  X  /\  w  e.  X )  ->  ( ( tail `  D
) `  w )  e.  ran  ( tail `  D
) )
3634, 35sylan 471 . . . . . . . . . . . 12  |-  ( ( D  e.  DirRel  /\  w  e.  X )  ->  (
( tail `  D ) `  w )  e.  ran  ( tail `  D )
)
3736ad2ant2r 746 . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . . . . . . . . 22  |-  x  e. 
_V
39 dirtr 15993 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( D  e.  DirRel  /\  x  e.  _V )  /\  ( u D w  /\  w D x ) )  ->  u D x )
4039exp32 605 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( D  e.  DirRel  /\  x  e.  _V )  ->  (
u D w  -> 
( w D x  ->  u D x ) ) )
4138, 40mpan2 671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  DirRel  ->  ( u D w  ->  ( w D x  ->  u D x ) ) )
4241com23 78 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  DirRel  ->  ( w D x  ->  ( u D w  ->  u D x ) ) )
4342imp 429 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  DirRel  /\  w D x )  -> 
( u D w  ->  u D x ) )
4443ad2ant2rl 748 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  w D x ) )  ->  ( u D w  ->  u D x ) )
45 dirtr 15993 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( D  e.  DirRel  /\  x  e.  _V )  /\  ( v D w  /\  w D x ) )  ->  v D x )
4645exp32 605 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( D  e.  DirRel  /\  x  e.  _V )  ->  (
v D w  -> 
( w D x  ->  v D x ) ) )
4738, 46mpan2 671 . . . . . . . . . . . . . . . . . . . . 21  |-  ( D  e.  DirRel  ->  ( v D w  ->  ( w D x  ->  v D x ) ) )
4847com23 78 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  DirRel  ->  ( w D x  ->  ( v D w  ->  v D x ) ) )
4948imp 429 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  DirRel  /\  w D x )  -> 
( v D w  ->  v D x ) )
5049ad2ant2rl 748 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( D  e.  DirRel  /\  ( u  e.  X  /\  v  e.  X
) )  /\  (
w  e.  X  /\  w D x ) )  ->  ( v D w  ->  v D x ) )
5144, 50anim12d 563 . . . . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . . . 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 619 . . . . . . . . . . . . . 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 30376 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  w  e.  X  /\  x  e.  _V )  ->  (
x  e.  ( (
tail `  D ) `  w )  <->  w D x ) )
5638, 55mp3an3 1313 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  DirRel  /\  w  e.  X )  ->  (
x  e.  ( (
tail `  D ) `  w )  <->  w D x ) )
5756ad2ant2r 746 . . . . . . . . . . . . . 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 30376 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  DirRel  /\  u  e.  X  /\  x  e.  _V )  ->  (
x  e.  ( (
tail `  D ) `  u )  <->  u D x ) )
5938, 58mp3an3 1313 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  DirRel  /\  u  e.  X )  ->  (
x  e.  ( (
tail `  D ) `  u )  <->  u D x ) )
6059adantrr 716 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  ( x  e.  ( ( tail `  D
) `  u )  <->  u D x ) )
611eltail 30376 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  DirRel  /\  v  e.  X  /\  x  e.  _V )  ->  (
x  e.  ( (
tail `  D ) `  v )  <->  v D x ) )
6238, 61mp3an3 1313 . . . . . . . . . . . . . . . . 17  |-  ( ( D  e.  DirRel  /\  v  e.  X )  ->  (
x  e.  ( (
tail `  D ) `  v )  <->  v D x ) )
6362adantrl 715 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  DirRel  /\  (
u  e.  X  /\  v  e.  X )
)  ->  ( x  e.  ( ( tail `  D
) `  v )  <->  v D x ) )
6460, 63anbi12d 710 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 268 . . . . . . . . . . . . 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 3683 . . . . . . . . . . . . 13  |-  ( x  e.  ( ( (
tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) )  <->  ( x  e.  ( ( tail `  D
) `  u )  /\  x  e.  (
( tail `  D ) `  v ) ) )
6866, 67syl6ibr 227 . . . . . . . . . . . 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 3505 . . . . . . . . . . 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 3520 . . . . . . . . . . . 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 3210 . . . . . . . . . . 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 661 . . . . . . . . . 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 2953 . . . . . . . . 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 3689 . . . . . . . . . . . 12  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( ( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
)  =  ( x  i^i  ( ( tail `  D ) `  v
) ) )
7574sseq2d 3527 . . . . . . . . . . 11  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( z 
C_  ( ( (
tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) )  <->  z  C_  ( x  i^i  (
( tail `  D ) `  v ) ) ) )
7675rexbidv 2968 . . . . . . . . . 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 3690 . . . . . . . . . . . 12  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( x  i^i  ( ( tail `  D ) `  v
) )  =  ( x  i^i  y ) )
7877sseq2d 3527 . . . . . . . . . . 11  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( z 
C_  ( x  i^i  ( ( tail `  D
) `  v )
)  <->  z  C_  (
x  i^i  y )
) )
7978rexbidv 2968 . . . . . . . . . 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 699 . . . . . . . . 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 220 . . . . . . . 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 2956 . . . . . . 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 218 . . . . . 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 215 . . . . 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 465 . . . 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 2877 . . 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 1176 . 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 6730 . . . . 5  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
891, 88syl5eqel 2549 . . . 4  |-  ( D  e.  DirRel  ->  X  e.  _V )
9089adantr 465 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  X  e.  _V )
91 isfbas2 20462 . . 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 922 1  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  ran  ( tail `  D )  e.  ( fBas `  X
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652    e/ wnel 2653   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   class class class wbr 4456   dom cdm 5008   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594   DirRelcdir 15985   tailctail 15986   fBascfbas 18533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-dir 15987  df-tail 15988  df-fbas 18543
This theorem is referenced by:  filnetlem4  30383
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