Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tailini Structured version   Visualization version   GIF version

Theorem tailini 36377
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 ∈ DirRel ∧ 𝐴𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴))

Proof of Theorem tailini
StepHypRef Expression
1 tailini.1 . . 3 𝑋 = dom 𝐷
21dirref 18646 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝐷𝐴)
31eltail 36375 . . 3 ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐴𝑋) → (𝐴 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐴))
433anidm23 1423 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → (𝐴 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐴))
52, 4mpbird 257 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  dom cdm 5685  cfv 6561  DirRelcdir 18639  tailctail 18640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-dir 18641  df-tail 18642
This theorem is referenced by:  tailfb  36378
  Copyright terms: Public domain W3C validator