Theorem iswwlksnon 26740

Description: 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
iswwlksnon.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
iswwlksnon	|- ( ( A e. V /\ B e. V ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )

Distinct variable groups:   w, A   w, B   w, G   w, N

Allowed substitution hint:   V( w)

Proof of Theorem iswwlksnon

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

Step	Hyp	Ref	Expression
1		iswwlksnon.v	. . . . . 6 |- V = (Vtx ` G )
2	1	wwlksnon 26738	. . . . 5 |- ( ( N e. NN0 /\ G e. _V ) -> ( N WWalksNOn G ) = ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) )
3	2	adantr 481	. . . 4 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) ) -> ( N WWalksNOn G ) = ( a e. V , b e. V |-> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } ) )
4		eqeq2 2633	. . . . . . 7 |- ( a = A -> ( ( w ` 0 ) = a <-> ( w ` 0 ) = A ) )
5		eqeq2 2633	. . . . . . 7 |- ( b = B -> ( ( w ` N ) = b <-> ( w ` N ) = B ) )
6	4, 5	bi2anan9 917	. . . . . 6 |- ( ( a = A /\ b = B ) -> ( ( ( w ` 0 ) = a /\ ( w ` N ) = b ) <-> ( ( w ` 0 ) = A /\ ( w ` N ) = B ) ) )
7	6	rabbidv 3189	. . . . 5 |- ( ( a = A /\ b = B ) -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )
8	7	adantl 482	. . . 4 |- ( ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) ) /\ ( a = A /\ b = B ) ) -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = a /\ ( w ` N ) = b ) } = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )
9		simprl 794	. . . 4 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) ) -> A e. V )
10		simprr 796	. . . 4 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) ) -> B e. V )
11		ovex 6678	. . . . . 6 |- ( N WWalksN G ) e. _V
12	11	rabex 4813	. . . . 5 |- { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } e. _V
13	12	a1i 11	. . . 4 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) ) -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } e. _V )
14	3, 8, 9, 10, 13	ovmpt2d 6788	. . 3 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )
15	14	ex 450	. 2 |- ( ( N e. NN0 /\ G e. _V ) -> ( ( A e. V /\ B e. V ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } ) )
16		0ov 6682	. . . 4 |- ( A (/) B ) = (/)
17		df-wwlksnon 26724	. . . . . 6 |- WWalksNOn = ( n e. NN0 , g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( n WWalksN g ) | ( ( w ` 0 ) = a /\ ( w ` n ) = b ) } ) )
18	17	mpt2ndm0 6875	. . . . 5 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( N WWalksNOn G ) = (/) )
19	18	oveqd 6667	. . . 4 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( A ( N WWalksNOn G ) B ) = ( A (/) B ) )
20		df-wwlksn 26723	. . . . . . 7 |- WWalksN = ( n e. NN0 , g e. _V |-> { w e. (WWalks ` g ) | ( # ` w ) = ( n + 1 ) } )
21	20	mpt2ndm0 6875	. . . . . 6 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( N WWalksN G ) = (/) )
22	21	rabeqdv 3194	. . . . 5 |- ( -. ( N e. NN0 /\ G e. _V ) -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } = { w e. (/) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )
23		rab0 3955	. . . . 5 |- { w e. (/) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } = (/)
24	22, 23	syl6eq 2672	. . . 4 |- ( -. ( N e. NN0 /\ G e. _V ) -> { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } = (/) )
25	16, 19, 24	3eqtr4a 2682	. . 3 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )
26	25	a1d 25	. 2 |- ( -. ( N e. NN0 /\ G e. _V ) -> ( ( A e. V /\ B e. V ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } ) )
27	15, 26	pm2.61i 176	1 |- ( ( A e. V /\ B e. V ) -> ( A ( N WWalksNOn G ) B ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = A /\ ( w ` N ) = B ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   #chash 13117  Vtxcvtx 25874  WWalkscwwlks 26717   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:  wwlknon  26742  wwlksnonfi  26816  wpthswwlks2on  26854