Theorem numclwwlkovh 27234

Description: Value of operation H, mapping a vertex v and a positive integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 30-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 ) ) } )

Assertion

Ref	Expression
numclwwlkovh	|- ( ( X e. V /\ N e. NN ) -> ( X H N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )

Distinct variable groups:   n, G, v, w   n, N, v, w   n, V, v   n, X, v, w   w, V

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

Proof of Theorem numclwwlkovh

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( n = N -> ( n ClWWalksN G ) = ( N ClWWalksN G ) )
2	1	adantl 482	. . 3 |- ( ( v = X /\ n = N ) -> ( n ClWWalksN G ) = ( N ClWWalksN G ) )
3		eqeq2 2633	. . . 4 |- ( v = X -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = X ) )
4		oveq1 6657	. . . . . 6 |- ( n = N -> ( n - 2 ) = ( N - 2 ) )
5	4	fveq2d 6195	. . . . 5 |- ( n = N -> ( w ` ( n - 2 ) ) = ( w ` ( N - 2 ) ) )
6	5	neeq1d 2853	. . . 4 |- ( n = N -> ( ( w ` ( n - 2 ) ) =/= ( w ` 0 ) <-> ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) )
7	3, 6	bi2anan9 917	. . 3 |- ( ( v = X /\ n = N ) -> ( ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) <-> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) ) )
8	2, 7	rabeqbidv 3195	. 2 |- ( ( v = X /\ n = N ) -> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )
9		numclwwlk.h	. 2 |- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
10		ovex 6678	. . 3 |- ( N ClWWalksN G ) e. _V
11	10	rabex 4813	. 2 |- { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } e. _V
12	8, 9, 11	ovmpt2a 6791	1 |- ( ( X e. V /\ N e. NN ) -> ( X H N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) =/= ( w ` 0 ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   - cmin 10266   NNcn 11020   2c2 11070   lastS clsw 13292  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-ne 2795  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-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  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  numclwwlk2lem1  27235  numclwlk2lem2f  27236  numclwlk2lem2f1o  27238  numclwwlk3lem  27241