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Theorem eltail 30167
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  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  A D B ) )

Proof of Theorem eltail
StepHypRef Expression
1 tailfval.1 . . . . 5  |-  X  =  dom  D
21tailval 30166 . . . 4  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
32eleq2d 2513 . . 3  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
433adant3 1017 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
5 elimasng 5353 . . . 4  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  <. A ,  B >.  e.  D ) )
6 df-br 4438 . . . 4  |-  ( A D B  <->  <. A ,  B >.  e.  D )
75, 6syl6bbr 263 . . 3  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  A D B ) )
873adant1 1015 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <-> 
A D B ) )
94, 8bitrd 253 1  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   {csn 4014   <.cop 4020   class class class wbr 4437   dom cdm 4989   "cima 4992   ` cfv 5578   DirRelcdir 15732   tailctail 15733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-dir 15734  df-tail 15735
This theorem is referenced by:  tailini  30169  tailfb  30170  filnetlem4  30174
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