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Theorem tailf 29647
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 5339 . . . . . . 7  |-  ( D
" { x }
)  C_  ran  D
2 ssun2 3661 . . . . . . . 8  |-  ran  D  C_  ( dom  D  u.  ran  D )
3 dmrnssfld 5252 . . . . . . . 8  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
42, 3sstri 3506 . . . . . . 7  |-  ran  D  C_ 
U. U. D
51, 4sstri 3506 . . . . . 6  |-  ( D
" { x }
)  C_  U. U. D
6 tailf.1 . . . . . . 7  |-  X  =  dom  D
7 dirdm 15710 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
86, 7syl5req 2514 . . . . . 6  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
95, 8syl5sseq 3545 . . . . 5  |-  ( D  e.  DirRel  ->  ( D " { x } ) 
C_  X )
10 dmexg 6705 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
116, 10syl5eqel 2552 . . . . . 6  |-  ( D  e.  DirRel  ->  X  e.  _V )
12 elpw2g 4603 . . . . . 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 2872 . . 3  |-  ( D  e.  DirRel  ->  A. x  e.  X  ( D " { x } )  e.  ~P X )
16 eqid 2460 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
1716fmpt 6033 . . 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 29644 . . 3  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
2019feq1d 5708 . 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 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    u. cun 3467    C_ wss 3469   ~Pcpw 4003   {csn 4020   U.cuni 4238    |-> cmpt 4498   dom cdm 4992   ran crn 4993   "cima 4995   -->wf 5575   ` cfv 5579   DirRelcdir 15704   tailctail 15705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-dir 15706  df-tail 15707
This theorem is referenced by:  tailfb  29649  filnetlem4  29653
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