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Theorem tailf 28734
Description: The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Hypothesis
Ref Expression
tailf.1  |-  X  =  dom  D
Assertion
Ref Expression
tailf  |-  ( D  e.  DirRel  ->  ( tail `  D
) : X --> ~P X
)

Proof of Theorem tailf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imassrn 5278 . . . . . . 7  |-  ( D
" { x }
)  C_  ran  D
2 ssun2 3618 . . . . . . . 8  |-  ran  D  C_  ( dom  D  u.  ran  D )
3 dmrnssfld 5196 . . . . . . . 8  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
42, 3sstri 3463 . . . . . . 7  |-  ran  D  C_ 
U. U. D
51, 4sstri 3463 . . . . . 6  |-  ( D
" { x }
)  C_  U. U. D
6 tailf.1 . . . . . . 7  |-  X  =  dom  D
7 dirdm 15506 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
86, 7syl5req 2505 . . . . . 6  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
95, 8syl5sseq 3502 . . . . 5  |-  ( D  e.  DirRel  ->  ( D " { x } ) 
C_  X )
10 dmexg 6609 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
116, 10syl5eqel 2543 . . . . . 6  |-  ( D  e.  DirRel  ->  X  e.  _V )
12 elpw2g 4553 . . . . . 6  |-  ( X  e.  _V  ->  (
( D " {
x } )  e. 
~P X  <->  ( D " { x } ) 
C_  X ) )
1311, 12syl 16 . . . . 5  |-  ( D  e.  DirRel  ->  ( ( D
" { x }
)  e.  ~P X  <->  ( D " { x } )  C_  X
) )
149, 13mpbird 232 . . . 4  |-  ( D  e.  DirRel  ->  ( D " { x } )  e.  ~P X )
1514ralrimivw 2823 . . 3  |-  ( D  e.  DirRel  ->  A. x  e.  X  ( D " { x } )  e.  ~P X )
16 eqid 2451 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
1716fmpt 5963 . . 3  |-  ( A. x  e.  X  ( D " { x }
)  e.  ~P X  <->  ( x  e.  X  |->  ( D " { x } ) ) : X --> ~P X )
1815, 17sylib 196 . 2  |-  ( D  e.  DirRel  ->  ( x  e.  X  |->  ( D " { x } ) ) : X --> ~P X
)
196tailfval 28731 . . 3  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
2019feq1d 5644 . 2  |-  ( D  e.  DirRel  ->  ( ( tail `  D ) : X --> ~P X  <->  ( x  e.  X  |->  ( D " { x } ) ) : X --> ~P X
) )
2118, 20mpbird 232 1  |-  ( D  e.  DirRel  ->  ( tail `  D
) : X --> ~P X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3068    u. cun 3424    C_ wss 3426   ~Pcpw 3958   {csn 3975   U.cuni 4189    |-> cmpt 4448   dom cdm 4938   ran crn 4939   "cima 4941   -->wf 5512   ` cfv 5516   DirRelcdir 15500   tailctail 15501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-dir 15502  df-tail 15503
This theorem is referenced by:  tailfb  28736  filnetlem4  28740
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