Theorem numclwwlkovgel 27221

Description: Properties of an element of the value of operation C. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypothesis

Ref	Expression
numclwwlkovg.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwwlkovgel	|- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( W e. ( X C 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, W

Allowed substitution hints:   C( w, v, n)   V( w)   W( v, n)

Proof of Theorem numclwwlkovgel

Step	Hyp	Ref	Expression
1		numclwwlkovg.c	. . . . 5 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
2	1	numclwwlkovg 27220	. . . 4 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
3	2	eleq2d 2687	. . 3 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( W e. ( X C N ) <-> W e. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } ) )
4		fveq1 6190	. . . . . 6 |- ( w = W -> ( w ` 0 ) = ( W ` 0 ) )
5	4	eqeq1d 2624	. . . . 5 |- ( w = W -> ( ( w ` 0 ) = X <-> ( W ` 0 ) = X ) )
6		fveq1 6190	. . . . . 6 |- ( w = W -> ( w ` ( N - 2 ) ) = ( W ` ( N - 2 ) ) )
7	6, 4	eqeq12d 2637	. . . . 5 |- ( w = W -> ( ( w ` ( N - 2 ) ) = ( w ` 0 ) <-> ( W ` ( N - 2 ) ) = ( W ` 0 ) ) )
8	5, 7	anbi12d 747	. . . 4 |- ( w = W -> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) <-> ( ( W ` 0 ) = X /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) ) )
9	8	elrab 3363	. . 3 |- ( W e. { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } <-> ( W e. ( N ClWWalksN G ) /\ ( ( W ` 0 ) = X /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) ) )
10	3, 9	syl6bb 276	. 2 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( W e. ( X C N ) <-> ( W e. ( N ClWWalksN G ) /\ ( ( W ` 0 ) = X /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) ) ) )
11		3anass 1042	. 2 |- ( ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) <-> ( W e. ( N ClWWalksN G ) /\ ( ( W ` 0 ) = X /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) ) )
12	10, 11	syl6bbr 278	1 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( W e. ( X C N ) <-> ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   - cmin 10266   2c2 11070   ZZ>=cuz 11687   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-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:  numclwlk1lem2f1  27227