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Theorem eltail 29782
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 29781 . . . 4  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
32eleq2d 2530 . . 3  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
433adant3 1011 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( ( tail `  D ) `  A )  <->  B  e.  ( D " { A } ) ) )
5 elimasng 5354 . . . 4  |-  ( ( A  e.  X  /\  B  e.  C )  ->  ( B  e.  ( D " { A } )  <->  <. A ,  B >.  e.  D ) )
6 df-br 4441 . . . 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 1009 . 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 968    = wceq 1374    e. wcel 1762   {csn 4020   <.cop 4026   class class class wbr 4440   dom cdm 4992   "cima 4995   ` cfv 5579   DirRelcdir 15704   tailctail 15705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-dir 15706  df-tail 15707
This theorem is referenced by:  tailini  29784  tailfb  29785  filnetlem4  29789
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