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Theorem tailini 26405
Description: A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
Hypothesis
Ref Expression
tailini.1  |-  X  =  dom  D
Assertion
Ref Expression
tailini  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  A  e.  ( ( tail `  D
) `  A )
)

Proof of Theorem tailini
StepHypRef Expression
1 tailini.1 . . 3  |-  X  =  dom  D
21dirref 14680 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  A D A )
31eltail 26403 . . 3  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  A  e.  X )  ->  ( A  e.  ( ( tail `  D ) `  A )  <->  A D A ) )
433anidm23 1243 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  ( A  e.  ( ( tail `  D ) `  A )  <->  A D A ) )
52, 4mpbird 224 1  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  A  e.  ( ( tail `  D
) `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   dom cdm 4878   ` cfv 5454   DirRelcdir 14673   tailctail 14674
This theorem is referenced by:  tailfb  26406
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-dir 14675  df-tail 14676
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