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Theorem tailval 26093
Description: The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Hypothesis
Ref Expression
tailfval.1  |-  X  =  dom  D
Assertion
Ref Expression
tailval  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )

Proof of Theorem tailval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tailfval.1 . . . . 5  |-  X  =  dom  D
21tailfval 26092 . . . 4  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
32fveq1d 5670 . . 3  |-  ( D  e.  DirRel  ->  ( ( tail `  D ) `  A
)  =  ( ( x  e.  X  |->  ( D " { x } ) ) `  A ) )
43adantr 452 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( ( x  e.  X  |->  ( D " {
x } ) ) `
 A ) )
5 id 20 . . 3  |-  ( A  e.  X  ->  A  e.  X )
6 imaexg 5157 . . 3  |-  ( D  e.  DirRel  ->  ( D " { A } )  e. 
_V )
7 sneq 3768 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
87imaeq2d 5143 . . . 4  |-  ( x  =  A  ->  ( D " { x }
)  =  ( D
" { A }
) )
9 eqid 2387 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
108, 9fvmptg 5743 . . 3  |-  ( ( A  e.  X  /\  ( D " { A } )  e.  _V )  ->  ( ( x  e.  X  |->  ( D
" { x }
) ) `  A
)  =  ( D
" { A }
) )
115, 6, 10syl2anr 465 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( D " {
x } ) ) `
 A )  =  ( D " { A } ) )
124, 11eqtrd 2419 1  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   {csn 3757    e. cmpt 4207   dom cdm 4818   "cima 4821   ` cfv 5394   DirRelcdir 14600   tailctail 14601
This theorem is referenced by:  eltail  26094
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-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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