Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tailfb Structured version   Unicode version

Theorem tailfb 30122
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 30120 . . . 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 3799 . . . . 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 3796 . . . . . . 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 1700 . . . . 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 30121 . . . . . . . 8  |-  ( ( D  e.  DirRel  /\  x  e.  X )  ->  x  e.  ( ( tail `  D
) `  x )
)
17 n0i 3795 . . . . . . . 8  |-  ( x  e.  ( ( tail `  D ) `  x
)  ->  -.  (
( tail `  D ) `  x )  =  (/) )
1816, 17syl 16 . . . . . . 7  |-  ( ( D  e.  DirRel  /\  x  e.  X )  ->  -.  ( ( tail `  D
) `  x )  =  (/) )
1918nrexdv 2923 . . . . . 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 5921 . . . . . . 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 2665 . . . 4  |-  ( (/)  e/ 
ran  ( tail `  D
)  <->  -.  (/)  e.  ran  ( tail `  D )
)
2624, 25sylibr 212 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  (/)  e/  ran  ( tail `  D )
)
27 fvelrnb 5921 . . . . . . . 8  |-  ( (
tail `  D )  Fn  X  ->  ( x  e.  ran  ( tail `  D )  <->  E. u  e.  X  ( ( tail `  D ) `  u )  =  x ) )
28 fvelrnb 5921 . . . . . . . 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 3034 . . . . . . 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 15741 . . . . . . . . . . 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 3121 . . . . . . . . . . . . . . . . . . . . . 22  |-  x  e. 
_V
39 dirtr 15740 . . . . . . . . . . . . . . . . . . . . . . 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 15740 . . . . . . . . . . . . . . . . . . . . . . 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 30119 . . . . . . . . . . . . . . . 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 30119 . . . . . . . . . . . . . . . . . 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 30119 . . . . . . . . . . . . . . . . . 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 3692 . . . . . . . . . . . . 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 3515 . . . . . . . . . . 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 3530 . . . . . . . . . . . 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 3219 . . . . . . . . . . 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 2963 . . . . . . . . 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 3698 . . . . . . . . . . . 12  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( ( ( tail `  D
) `  u )  i^i  ( ( tail `  D
) `  v )
)  =  ( x  i^i  ( ( tail `  D ) `  v
) ) )
7574sseq2d 3537 . . . . . . . . . . 11  |-  ( ( ( tail `  D
) `  u )  =  x  ->  ( z 
C_  ( ( (
tail `  D ) `  u )  i^i  (
( tail `  D ) `  v ) )  <->  z  C_  ( x  i^i  (
( tail `  D ) `  v ) ) ) )
7675rexbidv 2978 . . . . . . . . . 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 3699 . . . . . . . . . . . 12  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( x  i^i  ( ( tail `  D ) `  v
) )  =  ( x  i^i  y ) )
7877sseq2d 3537 . . . . . . . . . . 11  |-  ( ( ( tail `  D
) `  v )  =  y  ->  ( z 
C_  ( x  i^i  ( ( tail `  D
) `  v )
)  <->  z  C_  (
x  i^i  y )
) )
7978rexbidv 2978 . . . . . . . . . 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 2966 . . . . . . 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 2887 . . 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 6726 . . . . 5  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
891, 88syl5eqel 2559 . . . 4  |-  ( D  e.  DirRel  ->  X  e.  _V )
9089adantr 465 . . 3  |-  ( ( D  e.  DirRel  /\  X  =/=  (/) )  ->  X  e.  _V )
91 isfbas2 20204 . . 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 920 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662    e/ wnel 2663   A.wral 2817   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   class class class wbr 4453   dom cdm 5005   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594   DirRelcdir 15732   tailctail 15733   fBascfbas 18276
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 15734  df-tail 15735  df-fbas 18286
This theorem is referenced by:  filnetlem4  30126
  Copyright terms: Public domain W3C validator