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Theorem clwwlkfv 24499
Description: Lemma 2 for clwwlkbij 24503: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.)
Hypotheses
Ref Expression
clwwlkbij.d  |-  D  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( lastS  `  w )  =  ( w `  0 ) }
clwwlkbij.f  |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )
Assertion
Ref Expression
clwwlkfv  |-  ( W  e.  D  ->  ( F `  W )  =  ( W substr  <. 0 ,  N >. ) )
Distinct variable groups:    w, E    w, N    w, V    t, D    t, E, w    t, N    t, V    t, W
Allowed substitution hints:    D( w)    F( w, t)    W( w)

Proof of Theorem clwwlkfv
StepHypRef Expression
1 ovex 6309 . 2  |-  ( W substr  <. 0 ,  N >. )  e.  _V
2 oveq1 6291 . . 3  |-  ( t  =  W  ->  (
t substr  <. 0 ,  N >. )  =  ( W substr  <. 0 ,  N >. ) )
3 clwwlkbij.f . . 3  |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )
42, 3fvmptg 5948 . 2  |-  ( ( W  e.  D  /\  ( W substr  <. 0 ,  N >. )  e.  _V )  ->  ( F `  W )  =  ( W substr  <. 0 ,  N >. ) )
51, 4mpan2 671 1  |-  ( W  e.  D  ->  ( F `  W )  =  ( W substr  <. 0 ,  N >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113   <.cop 4033    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   0cc0 9492   lastS clsw 12501   substr csubstr 12504   WWalksN cwwlkn 24382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287
This theorem is referenced by:  clwwlkf1  24500  clwwlkfo  24501
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