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Theorem tailfval 31076
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 6614 . . . 4  |-  ( D  e.  DirRel  ->  U. D  e.  _V )
2 uniexg 6614 . . . 4  |-  ( U. D  e.  _V  ->  U.
U. D  e.  _V )
3 mptexg 6159 . . . 4  |-  ( U. U. D  e.  _V  ->  ( x  e.  U. U. D  |->  ( D " { x } ) )  e.  _V )
41, 2, 33syl 18 . . 3  |-  ( D  e.  DirRel  ->  ( x  e. 
U. U. D  |->  ( D
" { x }
) )  e.  _V )
5 unieq 4219 . . . . . 6  |-  ( d  =  D  ->  U. d  =  U. D )
65unieqd 4221 . . . . 5  |-  ( d  =  D  ->  U. U. d  =  U. U. D
)
7 imaeq1 5181 . . . . 5  |-  ( d  =  D  ->  (
d " { x } )  =  ( D " { x } ) )
86, 7mpteq12dv 4494 . . . 4  |-  ( d  =  D  ->  (
x  e.  U. U. d  |->  ( d " { x } ) )  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
9 df-tail 16525 . . . 4  |-  tail  =  ( d  e.  DirRel  |->  ( x  e.  U. U. d  |->  ( d " { x } ) ) )
108, 9fvmptg 5968 . . 3  |-  ( ( D  e.  DirRel  /\  (
x  e.  U. U. D  |->  ( D " { x } ) )  e.  _V )  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
114, 10mpdan 679 . 2  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
12 tailfval.1 . . . 4  |-  X  =  dom  D
13 dirdm 16528 . . . 4  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
1412, 13syl5req 2508 . . 3  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
1514mpteq1d 4497 . 2  |-  ( D  e.  DirRel  ->  ( x  e. 
U. U. D  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) ) )
1611, 15eqtrd 2495 1  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1454    e. wcel 1897   _Vcvv 3056   {csn 3979   U.cuni 4211    |-> cmpt 4474   dom cdm 4852   "cima 4855   ` cfv 5600   DirRelcdir 16522   tailctail 16523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-dir 16524  df-tail 16525
This theorem is referenced by:  tailval  31077  tailf  31079
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