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Theorem tailini 30604
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 16189 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  A D A )
31eltail 30602 . . 3  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  A  e.  X )  ->  ( A  e.  ( ( tail `  D ) `  A )  <->  A D A ) )
433anidm23 1289 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  ( A  e.  ( ( tail `  D ) `  A )  <->  A D A ) )
52, 4mpbird 232 1  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  A  e.  ( ( tail `  D
) `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4395   dom cdm 4823   ` 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-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
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