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Theorem tailf 26095
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 5156 . . . . . . 7  |-  ( D
" { x }
)  C_  ran  D
2 ssun2 3454 . . . . . . . 8  |-  ran  D  C_  ( dom  D  u.  ran  D )
3 dmrnssfld 5069 . . . . . . . 8  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
42, 3sstri 3300 . . . . . . 7  |-  ran  D  C_ 
U. U. D
51, 4sstri 3300 . . . . . 6  |-  ( D
" { x }
)  C_  U. U. D
6 tailf.1 . . . . . . 7  |-  X  =  dom  D
7 dirdm 14606 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
86, 7syl5req 2432 . . . . . 6  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
95, 8syl5sseq 3339 . . . . 5  |-  ( D  e.  DirRel  ->  ( D " { x } ) 
C_  X )
10 dmexg 5070 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
116, 10syl5eqel 2471 . . . . . 6  |-  ( D  e.  DirRel  ->  X  e.  _V )
12 elpw2g 4304 . . . . . 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 224 . . . 4  |-  ( D  e.  DirRel  ->  ( D " { x } )  e.  ~P X )
1514ralrimivw 2733 . . 3  |-  ( D  e.  DirRel  ->  A. x  e.  X  ( D " { x } )  e.  ~P X )
16 eqid 2387 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
1716fmpt 5829 . . 3  |-  ( A. x  e.  X  ( D " { x }
)  e.  ~P X  <->  ( x  e.  X  |->  ( D " { x } ) ) : X --> ~P X )
1815, 17sylib 189 . 2  |-  ( D  e.  DirRel  ->  ( x  e.  X  |->  ( D " { x } ) ) : X --> ~P X
)
196tailfval 26092 . . 3  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
2019feq1d 5520 . 2  |-  ( D  e.  DirRel  ->  ( ( tail `  D ) : X --> ~P X  <->  ( x  e.  X  |->  ( D " { x } ) ) : X --> ~P X
) )
2118, 20mpbird 224 1  |-  ( D  e.  DirRel  ->  ( tail `  D
) : X --> ~P X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    u. cun 3261    C_ wss 3263   ~Pcpw 3742   {csn 3757   U.cuni 3957    e. cmpt 4207   dom cdm 4818   ran crn 4819   "cima 4821   -->wf 5390   ` cfv 5394   DirRelcdir 14600   tailctail 14601
This theorem is referenced by:  tailfb  26097  filnetlem4  26101
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-dir 14602  df-tail 14603
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