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

Theorem tailval 26425
Description: The tail of an element in a directed set. (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
tailval  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )

Proof of Theorem tailval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tailfval.1 . . . . 5  |-  X  =  dom  D
21tailfval 26424 . . . 4  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
32fveq1d 5543 . . 3  |-  ( D  e.  DirRel  ->  ( ( tail `  D ) `  A
)  =  ( ( x  e.  X  |->  ( D " { x } ) ) `  A ) )
43adantr 451 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( ( x  e.  X  |->  ( D " {
x } ) ) `
 A ) )
5 id 19 . . 3  |-  ( A  e.  X  ->  A  e.  X )
6 imaexg 5042 . . 3  |-  ( D  e.  DirRel  ->  ( D " { A } )  e. 
_V )
7 sneq 3664 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
87imaeq2d 5028 . . . 4  |-  ( x  =  A  ->  ( D " { x }
)  =  ( D
" { A }
) )
9 eqid 2296 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
108, 9fvmptg 5616 . . 3  |-  ( ( A  e.  X  /\  ( D " { A } )  e.  _V )  ->  ( ( x  e.  X  |->  ( D
" { x }
) ) `  A
)  =  ( D
" { A }
) )
115, 6, 10syl2anr 464 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( D " {
x } ) ) `
 A )  =  ( D " { A } ) )
124, 11eqtrd 2328 1  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653    e. cmpt 4093   dom cdm 4705   "cima 4708   ` cfv 5271   DirRelcdir 14366   tailctail 14367
This theorem is referenced by:  eltail  26426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-dir 14368  df-tail 14369
  Copyright terms: Public domain W3C validator