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Theorem tailf 26385
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 5208 . . . . . . 7  |-  ( D
" { x }
)  C_  ran  D
2 ssun2 3503 . . . . . . . 8  |-  ran  D  C_  ( dom  D  u.  ran  D )
3 dmrnssfld 5121 . . . . . . . 8  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
42, 3sstri 3349 . . . . . . 7  |-  ran  D  C_ 
U. U. D
51, 4sstri 3349 . . . . . 6  |-  ( D
" { x }
)  C_  U. U. D
6 tailf.1 . . . . . . 7  |-  X  =  dom  D
7 dirdm 14671 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
86, 7syl5req 2480 . . . . . 6  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
95, 8syl5sseq 3388 . . . . 5  |-  ( D  e.  DirRel  ->  ( D " { x } ) 
C_  X )
10 dmexg 5122 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
116, 10syl5eqel 2519 . . . . . 6  |-  ( D  e.  DirRel  ->  X  e.  _V )
12 elpw2g 4355 . . . . . 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 2782 . . 3  |-  ( D  e.  DirRel  ->  A. x  e.  X  ( D " { x } )  e.  ~P X )
16 eqid 2435 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
1716fmpt 5882 . . 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 26382 . . 3  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
2019feq1d 5572 . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    u. cun 3310    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007    e. cmpt 4258   dom cdm 4870   ran crn 4871   "cima 4873   -->wf 5442   ` cfv 5446   DirRelcdir 14665   tailctail 14666
This theorem is referenced by:  tailfb  26387  filnetlem4  26391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-dir 14667  df-tail 14668
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