Theorem numclwlk1lem2fv 27226

Description: Value of the function T. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
extwwlkfab.v	|- V = (Vtx ` G )
extwwlkfab.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
extwwlkfab.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
numclwwlk.t	|- T = ( u e. ( X C N ) |-> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. )

Assertion

Ref	Expression
numclwlk1lem2fv	|- ( W e. ( X C N ) -> ( T ` W ) = <. ( W substr <. 0 , ( N - 2 ) >. ) , ( W ` ( N - 1 ) ) >. )

Distinct variable groups:   n, G, u, v, w   n, N, u, v, w   n, V, v, w   n, X, u, v, w   w, F   w, W   u, C   u, F   u, V   u, W

Allowed substitution hints:   C( w, v, n)   T( w, v, u, n)   F( v, n)   W( v, n)

Proof of Theorem numclwlk1lem2fv

Step	Hyp	Ref	Expression
1		oveq1 6657	. . 3 |- ( u = W -> ( u substr <. 0 , ( N - 2 ) >. ) = ( W substr <. 0 , ( N - 2 ) >. ) )
2		fveq1 6190	. . 3 |- ( u = W -> ( u ` ( N - 1 ) ) = ( W ` ( N - 1 ) ) )
3	1, 2	opeq12d 4410	. 2 |- ( u = W -> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. = <. ( W substr <. 0 , ( N - 2 ) >. ) , ( W ` ( N - 1 ) ) >. )
4		numclwwlk.t	. 2 |- T = ( u e. ( X C N ) |-> <. ( u substr <. 0 , ( N - 2 ) >. ) , ( u ` ( N - 1 ) ) >. )
5		opex 4932	. 2 |- <. ( W substr <. 0 , ( N - 2 ) >. ) , ( W ` ( N - 1 ) ) >. e. _V
6	3, 4, 5	fvmpt 6282	1 |- ( W e. ( X C N ) -> ( T ` W ) = <. ( W substr <. 0 , ( N - 2 ) >. ) , ( W ` ( N - 1 ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   2c2 11070   ZZ>=cuz 11687   substr csubstr 13295  Vtxcvtx 25874   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-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:  numclwlk1lem2f1  27227  numclwlk1lem2fo  27228