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Theorem tailini 35893
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 ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((tailβ€˜π·)β€˜π΄))

Proof of Theorem tailini
StepHypRef Expression
1 tailini.1 . . 3 𝑋 = dom 𝐷
21dirref 18600 . 2 ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ 𝐴𝐷𝐴)
31eltail 35891 . . 3 ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ ((tailβ€˜π·)β€˜π΄) ↔ 𝐴𝐷𝐴))
433anidm23 1418 . 2 ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ ((tailβ€˜π·)β€˜π΄) ↔ 𝐴𝐷𝐴))
52, 4mpbird 256 1 ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((tailβ€˜π·)β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  dom cdm 5682  β€˜cfv 6553  DirRelcdir 18593  tailctail 18594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-dir 18595  df-tail 18596
This theorem is referenced by:  tailfb  35894
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