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Theorem clwwlkfv 30600
Description: Lemma 2 for clwwlkbij 30604: 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 6220 . 2  |-  ( W substr  <. 0 ,  N >. )  e.  _V
2 oveq1 6202 . . 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 5876 . 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 1370    e. wcel 1758   {crab 2800   _Vcvv 3072   <.cop 3986    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   0cc0 9388   lastS clsw 12335   substr csubstr 12338   WWalksN cwwlkn 30455
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198
This theorem is referenced by:  clwwlkf1  30601  clwwlkfo  30602
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