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

Theorem eltail 26096
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 26095 . . . 4  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
32eleq2d 2456 . . 3  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
433adant3 977 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
5 elimasng 5172 . . . 4  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  <. A ,  B >.  e.  D ) )
6 df-br 4156 . . . 4  |-  ( A D B  <->  <. A ,  B >.  e.  D )
75, 6syl6bbr 255 . . 3  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  A D B ) )
873adant1 975 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <-> 
A D B ) )
94, 8bitrd 245 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {csn 3759   <.cop 3762   class class class wbr 4155   dom cdm 4820   "cima 4823   ` cfv 5396   DirRelcdir 14602   tailctail 14603
This theorem is referenced by:  tailini  26098  tailfb  26099  filnetlem4  26103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-dir 14604  df-tail 14605
  Copyright terms: Public domain W3C validator