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Theorem tailval 28743
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 28742 . . . 4  |-  ( D  e.  DirRel  ->  ( tail `  D
)  =  ( x  e.  X  |->  ( D
" { x }
) ) )
32fveq1d 5802 . . 3  |-  ( D  e.  DirRel  ->  ( ( tail `  D ) `  A
)  =  ( ( x  e.  X  |->  ( D " { x } ) ) `  A ) )
43adantr 465 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( ( x  e.  X  |->  ( D " {
x } ) ) `
 A ) )
5 id 22 . . 3  |-  ( A  e.  X  ->  A  e.  X )
6 imaexg 6626 . . 3  |-  ( D  e.  DirRel  ->  ( D " { A } )  e. 
_V )
7 sneq 3996 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
87imaeq2d 5278 . . . 4  |-  ( x  =  A  ->  ( D " { x }
)  =  ( D
" { A }
) )
9 eqid 2454 . . . 4  |-  ( x  e.  X  |->  ( D
" { x }
) )  =  ( x  e.  X  |->  ( D " { x } ) )
108, 9fvmptg 5882 . . 3  |-  ( ( A  e.  X  /\  ( D " { A } )  e.  _V )  ->  ( ( x  e.  X  |->  ( D
" { x }
) ) `  A
)  =  ( D
" { A }
) )
115, 6, 10syl2anr 478 . 2  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( D " {
x } ) ) `
 A )  =  ( D " { A } ) )
124, 11eqtrd 2495 1  |-  ( ( D  e.  DirRel  /\  A  e.  X )  ->  (
( tail `  D ) `  A )  =  ( D " { A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   {csn 3986    |-> cmpt 4459   dom cdm 4949   "cima 4952   ` cfv 5527   DirRelcdir 15518   tailctail 15519
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-dir 15520  df-tail 15521
This theorem is referenced by:  eltail  28744
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