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Theorem tailfval 28717
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
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 uniexg 6463 . . . 4  |-  ( D  e.  DirRel  ->  U. D  e.  _V )
2 uniexg 6463 . . . 4  |-  ( U. D  e.  _V  ->  U.
U. D  e.  _V )
3 mptexg 6032 . . . 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 4183 . . . . . 6  |-  ( d  =  D  ->  U. d  =  U. D )
65unieqd 4185 . . . . 5  |-  ( d  =  D  ->  U. U. d  =  U. U. D
)
7 imaeq1 5248 . . . . 5  |-  ( d  =  D  ->  (
d " { x } )  =  ( D " { x } ) )
86, 7mpteq12dv 4454 . . . 4  |-  ( d  =  D  ->  (
x  e.  U. U. d  |->  ( d " { x } ) )  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
9 df-tail 15489 . . . 4  |-  tail  =  ( d  e.  DirRel  |->  ( x  e.  U. U. d  |->  ( d " { x } ) ) )
108, 9fvmptg 5857 . . 3  |-  ( ( D  e.  DirRel  /\  (
x  e.  U. U. D  |->  ( D " { x } ) )  e.  _V )  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
114, 10mpdan 668 . 2  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
12 tailfval.1 . . . 4  |-  X  =  dom  D
13 dirdm 15492 . . . 4  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
1412, 13syl5req 2503 . . 3  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
1514mpteq1d 4457 . 2  |-  ( D  e.  DirRel  ->  ( x  e. 
U. U. D  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) ) )
1611, 15eqtrd 2490 1  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1757   _Vcvv 3054   {csn 3961   U.cuni 4175    |-> cmpt 4434   dom cdm 4924   "cima 4927   ` cfv 5502   DirRelcdir 15486   tailctail 15487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-dir 15488  df-tail 15489
This theorem is referenced by:  tailval  28718  tailf  28720
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