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Theorem clwwlkfv 25368
Description: Lemma 2 for clwwlkbij 25372: 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 oveq1 6312 . 2  |-  ( t  =  W  ->  (
t substr  <. 0 ,  N >. )  =  ( W substr  <. 0 ,  N >. ) )
2 clwwlkbij.f . 2  |-  F  =  ( t  e.  D  |->  ( t substr  <. 0 ,  N >. ) )
3 ovex 6333 . 2  |-  ( W substr  <. 0 ,  N >. )  e.  _V
41, 2, 3fvmpt 5964 1  |-  ( W  e.  D  ->  ( F `  W )  =  ( W substr  <. 0 ,  N >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   {crab 2786   <.cop 4008    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   0cc0 9538   lastS clsw 12644   substr csubstr 12647   WWalksN cwwlkn 25251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308
This theorem is referenced by:  clwwlkf1  25369  clwwlkfo  25370
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