Theorem iswwlksn 26730

Description: A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)

Assertion

Ref	Expression
iswwlksn	|- ( N e. NN0 -> ( W e. ( N WWalksN G ) <-> ( W e. (WWalks ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) )

Proof of Theorem iswwlksn

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksn 26729	. . 3 |- ( N e. NN0 -> ( N WWalksN G ) = { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } )
2	1	eleq2d 2687	. 2 |- ( N e. NN0 -> ( W e. ( N WWalksN G ) <-> W e. { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } ) )
3		fveq2 6191	. . . 4 |- ( w = W -> ( # ` w ) = ( # ` W ) )
4	3	eqeq1d 2624	. . 3 |- ( w = W -> ( ( # ` w ) = ( N + 1 ) <-> ( # ` W ) = ( N + 1 ) ) )
5	4	elrab 3363	. 2 |- ( W e. { w e. (WWalks ` G ) | ( # ` w ) = ( N + 1 ) } <-> ( W e. (WWalks ` G ) /\ ( # ` W ) = ( N + 1 ) ) )
6	2, 5	syl6bb 276	1 |- ( N e. NN0 -> ( W e. ( N WWalksN G ) <-> ( W e. (WWalks ` G ) /\ ( # ` W ) = ( N + 1 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   NN0cn0 11292   #chash 13117  WWalkscwwlks 26717   WWalksN cwwlksn 26718

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-sep 4781  ax-nul 4789  ax-pow 4843  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  df-wwlksn 26723

This theorem is referenced by:  iswwlksnx  26731  wwlknbp  26733  wwlknp  26734  wwlkswwlksn  26750  wlklnwwlkln1  26754  wlklnwwlkln2lem  26768  wlknewwlksn  26773  wwlksnred  26787  wwlksnext  26788  wwlksnextproplem3  26806  wspthsnonn0vne  26813  elwspths2spth  26862  rusgrnumwwlkl1  26863  clwwlksel  26914  clwwlksf  26915