Theorem wwlknon 26742

Description: An element of the set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)

Hypothesis

Ref	Expression
wwlknon.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
wwlknon	|- ( ( A e. V /\ B e. V ) -> ( W e. ( A ( N WWalksNOn G ) B ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` N ) = B ) ) )

Proof of Theorem wwlknon

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknon.v	. . . 4 |- V = (Vtx ` G )
2	1	iswwlksnon 26740	. . 3 |- ( ( A e. V /\ B e. V ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )
3	2	eleq2d 2687	. 2 |- ( ( A e. V /\ B e. V ) -> ( W e. ( A ( N WWalksNOn G ) B ) <-> W e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } ) )
4		fveq1 6190	. . . . . 6 |- ( w = W -> ( w ` 0 ) = ( W ` 0 ) )
5	4	eqeq1d 2624	. . . . 5 |- ( w = W -> ( ( w ` 0 ) = A <-> ( W ` 0 ) = A ) )
6		fveq1 6190	. . . . . 6 |- ( w = W -> ( w ` N ) = ( W ` N ) )
7	6	eqeq1d 2624	. . . . 5 |- ( w = W -> ( ( w ` N ) = B <-> ( W ` N ) = B ) )
8	5, 7	anbi12d 747	. . . 4 |- ( w = W -> ( ( ( w ` 0 ) = A /\ ( w ` N ) = B ) <-> ( ( W ` 0 ) = A /\ ( W ` N ) = B ) ) )
9	8	elrab 3363	. . 3 |- ( W e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } <-> ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = A /\ ( W ` N ) = B ) ) )
10		3anass 1042	. . 3 |- ( ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` N ) = B ) <-> ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = A /\ ( W ` N ) = B ) ) )
11	9, 10	bitr4i 267	. 2 |- ( W e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` N ) = B ) )
12	3, 11	syl6bb 276	1 |- ( ( A e. V /\ B e. V ) -> ( W e. ( A ( N WWalksNOn G ) B ) <-> ( W e. ( N WWalksN G ) /\ ( W ` 0 ) = A /\ ( W ` N ) = B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888  (class class class)co 6650   0cc0 9936  Vtxcvtx 25874   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  wwlksnwwlksnon  26810  wspthsnwspthsnon  26811  wspthsnonn0vne  26813  elwwlks2ons3  26848  s3wwlks2on  26849  wpthswwlks2on  26854  elwspths2spth  26862