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Theorem tailf 30603
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 5168 . . . . . . 7  |-  ( D
" { x }
)  C_  ran  D
2 ssun2 3607 . . . . . . . 8  |-  ran  D  C_  ( dom  D  u.  ran  D )
3 dmrnssfld 5082 . . . . . . . 8  |-  ( dom 
D  u.  ran  D
)  C_  U. U. D
42, 3sstri 3451 . . . . . . 7  |-  ran  D  C_ 
U. U. D
51, 4sstri 3451 . . . . . 6  |-  ( D
" { x }
)  C_  U. U. D
6 tailf.1 . . . . . . 7  |-  X  =  dom  D
7 dirdm 16188 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
86, 7syl5req 2456 . . . . . 6  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
95, 8syl5sseq 3490 . . . . 5  |-  ( D  e.  DirRel  ->  ( D " { x } ) 
C_  X )
10 dmexg 6715 . . . . . . 7  |-  ( D  e.  DirRel  ->  dom  D  e.  _V )
116, 10syl5eqel 2494 . . . . . 6  |-  ( D  e.  DirRel  ->  X  e.  _V )
12 elpw2g 4557 . . . . . 6  |-  ( X  e.  _V  ->  (
( D " {
x } )  e. 
~P X  <->  ( D " { x } ) 
C_  X ) )
1311, 12syl 17 . . . . 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 2819 . . 3  |-  ( D  e.  DirRel  ->  A. x  e.  X  ( D " { x } )  e.  ~P X )
16 eqid 2402 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
1716fmpt 6030 . . 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 30600 . . 3  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
2019feq1d 5700 . 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 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059    u. cun 3412    C_ wss 3414   ~Pcpw 3955   {csn 3972   U.cuni 4191    |-> cmpt 4453   dom cdm 4823   ran crn 4824   "cima 4826   -->wf 5565   ` cfv 5569   DirRelcdir 16182   tailctail 16183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-dir 16184  df-tail 16185
This theorem is referenced by:  tailfb  30605  filnetlem4  30609
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