Theorem numclwlk2lem2fv 27237

Description: Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.)

Hypotheses

Ref	Expression
numclwwlk.v	|- V = (Vtx ` G )
numclwwlk.q	|- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
numclwwlk.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
numclwwlk.h	|- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
numclwwlk.r	|- R = ( x e. ( X H ( N + 2 ) ) |-> ( x substr <. 0 , ( N + 1 ) >. ) )

Assertion

Ref	Expression
numclwlk2lem2fv	|- ( ( X e. V /\ N e. NN ) -> ( W e. ( X H ( N + 2 ) ) -> ( R ` W ) = ( W substr <. 0 , ( N + 1 ) >. ) ) )

Distinct variable groups:   n, G, v, w   n, N, v, w   n, V, v   n, X, v, w   w, V   v, W, w   x, G, w   x, H   x, N   x, Q   x, V   x, X   x, W

Allowed substitution hints:   Q( w, v, n)   R( x, w, v, n)   F( x, w, v, n)   H( w, v, n)   W( n)

Proof of Theorem numclwlk2lem2fv

Step	Hyp	Ref	Expression
1		numclwwlk.r	. . . 4 |- R = ( x e. ( X H ( N + 2 ) ) |-> ( x substr <. 0 , ( N + 1 ) >. ) )
2	1	a1i 11	. . 3 |- ( ( ( X e. V /\ N e. NN ) /\ W e. ( X H ( N + 2 ) ) ) -> R = ( x e. ( X H ( N + 2 ) ) |-> ( x substr <. 0 , ( N + 1 ) >. ) ) )
3		oveq1 6657	. . . 4 |- ( x = W -> ( x substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) )
4	3	adantl 482	. . 3 |- ( ( ( ( X e. V /\ N e. NN ) /\ W e. ( X H ( N + 2 ) ) ) /\ x = W ) -> ( x substr <. 0 , ( N + 1 ) >. ) = ( W substr <. 0 , ( N + 1 ) >. ) )
5		simpr 477	. . 3 |- ( ( ( X e. V /\ N e. NN ) /\ W e. ( X H ( N + 2 ) ) ) -> W e. ( X H ( N + 2 ) ) )
6		ovexd 6680	. . 3 |- ( ( ( X e. V /\ N e. NN ) /\ W e. ( X H ( N + 2 ) ) ) -> ( W substr <. 0 , ( N + 1 ) >. ) e. _V )
7	2, 4, 5, 6	fvmptd 6288	. 2 |- ( ( ( X e. V /\ N e. NN ) /\ W e. ( X H ( N + 2 ) ) ) -> ( R ` W ) = ( W substr <. 0 , ( N + 1 ) >. ) )
8	7	ex 450	1 |- ( ( X e. V /\ N e. NN ) -> ( W e. ( X H ( N + 2 ) ) -> ( R ` W ) = ( W substr <. 0 , ( N + 1 ) >. ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876

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-csb 3534  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:  numclwlk2lem2f1o  27238