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

Theorem eltail 26323
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 26322 . . . 4  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
32eleq2d 2350 . . 3  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
433adant3 975 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
5 elimasng 5039 . . . 4  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  <. A ,  B >.  e.  D ) )
6 df-br 4024 . . . 4  |-  ( A D B  <->  <. A ,  B >.  e.  D )
75, 6syl6bbr 254 . . 3  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  A D B ) )
873adant1 973 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <-> 
A D B ) )
94, 8bitrd 244 1  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  A D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   class class class wbr 4023   dom cdm 4689   "cima 4692   ` cfv 5255   DirRelcdir 14350   tailctail 14351
This theorem is referenced by:  tailini  26325  tailfb  26326  filnetlem4  26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dir 14352  df-tail 14353
  Copyright terms: Public domain W3C validator