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Theorem tailval 26322
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 26321 . . . 4  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
32fveq1d 5527 . . 3  |-  ( D  e.  DirRel  ->  ( ( tail `  D ) `  A
)  =  ( ( x  e.  X  |->  ( D " { x } ) ) `  A ) )
43adantr 451 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( ( x  e.  X  |->  ( D " {
x } ) ) `
 A ) )
5 id 19 . . 3  |-  ( A  e.  X  ->  A  e.  X )
6 imaexg 5026 . . 3  |-  ( D  e.  DirRel  ->  ( D " { A } )  e. 
_V )
7 sneq 3651 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
87imaeq2d 5012 . . . 4  |-  ( x  =  A  ->  ( D " { x }
)  =  ( D
" { A }
) )
9 eqid 2283 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
108, 9fvmptg 5600 . . 3  |-  ( ( A  e.  X  /\  ( D " { A } )  e.  _V )  ->  ( ( x  e.  X  |->  ( D
" { x }
) ) `  A
)  =  ( D
" { A }
) )
115, 6, 10syl2anr 464 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( D " {
x } ) ) `
 A )  =  ( D " { A } ) )
124, 11eqtrd 2315 1  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077   dom cdm 4689   "cima 4692   ` cfv 5255   DirRelcdir 14350   tailctail 14351
This theorem is referenced by:  eltail  26323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dir 14352  df-tail 14353
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