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Theorem tailini 32346
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 17216 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝐷𝐴)
31eltail 32344 . . 3 ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐴𝑋) → (𝐴 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐴))
433anidm23 1383 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → (𝐴 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐴))
52, 4mpbird 247 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → 𝐴 ∈ ((tail‘𝐷)‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988   class class class wbr 4644  dom cdm 5104  cfv 5876  DirRelcdir 17209  tailctail 17210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-dir 17211  df-tail 17212
This theorem is referenced by:  tailfb  32347
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