Theorem wspthsn 26735

Description: The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)

Assertion

Ref	Expression
wspthsn	|- ( N WSPathsN G ) = { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w }

Distinct variable groups:   f, G, w   w, N

Allowed substitution hint:   N( f)

Proof of Theorem wspthsn

Dummy variables g n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 6659	. . . 4 |- ( ( n = N /\ g = G ) -> ( n WWalksN g ) = ( N WWalksN G ) )
2		fveq2 6191	. . . . . . 7 |- ( g = G -> (SPaths ` g ) = (SPaths ` G ) )
3	2	breqd 4664	. . . . . 6 |- ( g = G -> ( f (SPaths ` g ) w <-> f (SPaths ` G ) w ) )
4	3	exbidv 1850	. . . . 5 |- ( g = G -> ( E. f f (SPaths ` g ) w <-> E. f f (SPaths ` G ) w ) )
5	4	adantl 482	. . . 4 |- ( ( n = N /\ g = G ) -> ( E. f f (SPaths ` g ) w <-> E. f f (SPaths ` G ) w ) )
6	1, 5	rabeqbidv 3195	. . 3 |- ( ( n = N /\ g = G ) -> { w e. ( n WWalksN g ) | E. f f (SPaths ` g ) w } = { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w } )
7		df-wspthsn 26725	. . 3 |- WSPathsN = ( n e. NN0 , g e. _V |-> { w e. ( n WWalksN g ) | E. f f (SPaths ` g ) w } )
8		ovex 6678	. . . 4 |- ( N WWalksN G ) e. _V
9	8	rabex 4813	. . 3 |- { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w } e. _V
10	6, 7, 9	ovmpt2a 6791	. 2 |- ( ( N e. NN0 /\ G e. _V ) -> ( N WSPathsN G ) = { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w } )
11	7	mpt2ndm0 6875	. . 3 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( N WSPathsN G ) = (/) )
12		df-wwlksn 26723	. . . . . 6 |- WWalksN = ( n e. NN0 , g e. _V |-> { w e. (WWalks ` g ) | ( # ` w ) = ( n + 1 ) } )
13	12	mpt2ndm0 6875	. . . . 5 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( N WWalksN G ) = (/) )
14	13	rabeqdv 3194	. . . 4 |- ( -. ( N e. NN0 /\ G e. _V ) -> { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w } = { w e. (/) | E. f f (SPaths ` G ) w } )
15		rab0 3955	. . . 4 |- { w e. (/) | E. f f (SPaths ` G ) w } = (/)
16	14, 15	syl6eq 2672	. . 3 |- ( -. ( N e. NN0 /\ G e. _V ) -> { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w } = (/) )
17	11, 16	eqtr4d 2659	. 2 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( N WSPathsN G ) = { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w } )
18	10, 17	pm2.61i 176	1 |- ( N WSPathsN G ) = { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w }

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   NN0cn0 11292   #chash 13117  SPathscspths 26609  WWalkscwwlks 26717   WWalksN cwwlksn 26718   WSPathsN cwwspthsn 26720

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  df-wspthsn 26725

This theorem is referenced by:  iswspthn  26736  wspn0  26820