Theorem clwwlksfv 26916

Description: Lemma 2 for clwwlksbij 26920: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.)

Hypotheses

Ref	Expression
clwwlksbij.d	|- D = { w e. ( N WWalksN G ) | ( lastS ` w ) = ( w ` 0 ) }
clwwlksbij.f	|- F = ( t e. D |-> ( t substr <. 0 , N >. ) )

Assertion

Ref	Expression
clwwlksfv	|- ( W e. D -> ( F ` W ) = ( W substr <. 0 , N >. ) )

Distinct variable groups:   w, G   w, N   t, D   t, G, w   t, N   t, W

Allowed substitution hints:   D( w)   F( w, t)   W( w)

Proof of Theorem clwwlksfv

Step	Hyp	Ref	Expression
1		oveq1 6657	. 2 |- ( t = W -> ( t substr <. 0 , N >. ) = ( W substr <. 0 , N >. ) )
2		clwwlksbij.f	. 2 |- F = ( t e. D |-> ( t substr <. 0 , N >. ) )
3		ovex 6678	. 2 |- ( W substr <. 0 , N >. ) e. _V
4	1, 2, 3	fvmpt 6282	1 |- ( W e. D -> ( F ` W ) = ( W substr <. 0 , N >. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   0cc0 9936   lastS clsw 13292   substr csubstr 13295   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  clwwlksf1  26917  clwwlksfo  26918