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

Theorem tailfval 25674
Description: The tail function for 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
tailfval  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
Distinct variable groups:    x, D    x, X

Proof of Theorem tailfval
StepHypRef Expression
1 uniexg 4475 . . . 4  |-  ( D  e.  DirRel  ->  U. D  e.  _V )
2 uniexg 4475 . . . 4  |-  ( U. D  e.  _V  ->  U.
U. D  e.  _V )
3 mptexg 5665 . . . 4  |-  ( U. U. D  e.  _V  ->  ( x  e.  U. U. D  |->  ( D " { x } ) )  e.  _V )
41, 2, 33syl 20 . . 3  |-  ( D  e.  DirRel  ->  ( x  e. 
U. U. D  |->  ( D
" { x }
) )  e.  _V )
5 unieq 3796 . . . . . 6  |-  ( d  =  D  ->  U. d  =  U. D )
65unieqd 3798 . . . . 5  |-  ( d  =  D  ->  U. U. d  =  U. U. D
)
7 imaeq1 4981 . . . . 5  |-  ( d  =  D  ->  (
d " { x } )  =  ( D " { x } ) )
86, 7mpteq12dv 4058 . . . 4  |-  ( d  =  D  ->  (
x  e.  U. U. d  |->  ( d " { x } ) )  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
9 df-tail 14301 . . . 4  |-  tail  =  ( d  e.  DirRel  |->  ( x  e.  U. U. d  |->  ( d " { x } ) ) )
108, 9fvmptg 5520 . . 3  |-  ( ( D  e.  DirRel  /\  (
x  e.  U. U. D  |->  ( D " { x } ) )  e.  _V )  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
114, 10mpdan 652 . 2  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
12 tailfval.1 . . . 4  |-  X  =  dom  D
13 dirdm 14304 . . . 4  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
1412, 13syl5req 2301 . . 3  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
15 eqidd 2257 . . 3  |-  ( D  e.  DirRel  ->  ( D " { x } )  =  ( D " { x } ) )
1614, 15mpteq12dv 4058 . 2  |-  ( D  e.  DirRel  ->  ( x  e. 
U. U. D  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) ) )
1711, 16eqtrd 2288 1  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   _Vcvv 2757   {csn 3600   U.cuni 3787    e. cmpt 4037   dom cdm 4647   "cima 4650   ` cfv 4659   DirRelcdir 14298   tailctail 14299
This theorem is referenced by:  tailval  25675  tailf  25677
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-dir 14300  df-tail 14301
  Copyright terms: Public domain W3C validator