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

Theorem eltail 32494
Description: An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Hypothesis
Ref Expression
tailfval.1 𝑋 = dom 𝐷
Assertion
Ref Expression
eltail ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐵𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐵))

Proof of Theorem eltail
StepHypRef Expression
1 tailfval.1 . . . . 5 𝑋 = dom 𝐷
21tailval 32493 . . . 4 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴}))
32eleq2d 2716 . . 3 ((𝐷 ∈ DirRel ∧ 𝐴𝑋) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐵 ∈ (𝐷 “ {𝐴})))
433adant3 1101 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐵𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐵 ∈ (𝐷 “ {𝐴})))
5 elimasng 5526 . . . 4 ((𝐴𝑋𝐵𝐶) → (𝐵 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐷))
6 df-br 4686 . . . 4 (𝐴𝐷𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐷)
75, 6syl6bbr 278 . . 3 ((𝐴𝑋𝐵𝐶) → (𝐵 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝐵))
873adant1 1099 . 2 ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐵𝐶) → (𝐵 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝐵))
94, 8bitrd 268 1 ((𝐷 ∈ DirRel ∧ 𝐴𝑋𝐵𝐶) → (𝐵 ∈ ((tail‘𝐷)‘𝐴) ↔ 𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {csn 4210  cop 4216   class class class wbr 4685  dom cdm 5143  cima 5146  cfv 5926  DirRelcdir 17275  tailctail 17276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-dir 17277  df-tail 17278
This theorem is referenced by:  tailini  32496  tailfb  32497  filnetlem4  32501
  Copyright terms: Public domain W3C validator