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Theorem tailfval 25721
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
Dummy variable  d is distinct from all other variables.

Proof of Theorem tailfval
StepHypRef Expression
1 uniexg 4517 . . . 4  |-  ( D  e.  DirRel  ->  U. D  e.  _V )
2 uniexg 4517 . . . 4  |-  ( U. D  e.  _V  ->  U.
U. D  e.  _V )
3 mptexg 5707 . . . 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 3838 . . . . . 6  |-  ( d  =  D  ->  U. d  =  U. D )
65unieqd 3840 . . . . 5  |-  ( d  =  D  ->  U. U. d  =  U. U. D
)
7 imaeq1 5007 . . . . 5  |-  ( d  =  D  ->  (
d " { x } )  =  ( D " { x } ) )
86, 7mpteq12dv 4100 . . . 4  |-  ( d  =  D  ->  (
x  e.  U. U. d  |->  ( d " { x } ) )  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
9 df-tail 14348 . . . 4  |-  tail  =  ( d  e.  DirRel  |->  ( x  e.  U. U. d  |->  ( d " { x } ) ) )
108, 9fvmptg 5562 . . 3  |-  ( ( D  e.  DirRel  /\  (
x  e.  U. U. D  |->  ( D " { x } ) )  e.  _V )  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
114, 10mpdan 651 . 2  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  U. U. D  |->  ( D " {
x } ) ) )
12 tailfval.1 . . . 4  |-  X  =  dom  D
13 dirdm 14351 . . . 4  |-  ( D  e.  DirRel  ->  dom  D  =  U. U. D )
1412, 13syl5req 2330 . . 3  |-  ( D  e.  DirRel  ->  U. U. D  =  X )
15 eqidd 2286 . . 3  |-  ( D  e.  DirRel  ->  ( D " { x } )  =  ( D " { x } ) )
1614, 15mpteq12dv 4100 . 2  |-  ( D  e.  DirRel  ->  ( x  e. 
U. U. D  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) ) )
1711, 16eqtrd 2317 1  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   _Vcvv 2790   {csn 3642   U.cuni 3829    e. cmpt 4079   dom cdm 4689   "cima 4692   ` cfv 5222   DirRelcdir 14345   tailctail 14346
This theorem is referenced by:  tailval  25722  tailf  25724
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-dir 14347  df-tail 14348
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