Theorem wspthnonp 26744

Description: Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.)

Hypothesis

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

Assertion

Ref	Expression
wspthnonp	|- ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) /\ ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) ) )

Distinct variable groups:   A, f   B, f   f, G   f, N   f, W

Allowed substitution hint:   V( f)

Proof of Theorem wspthnonp

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

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 |- (Vtx ` g ) e. _V
2	1, 1	pm3.2i 471	. . . 4 |- ( (Vtx ` g ) e. _V /\ (Vtx ` g ) e. _V )
3	2	rgen2w 2925	. . 3 |- A. n e. NN0 A. g e. _V ( (Vtx ` g ) e. _V /\ (Vtx ` g ) e. _V )
4		df-wspthsnon 26726	. . . 4 |- WSPathsNOn = ( n e. NN0 , g e. _V |-> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a (SPathsOn ` g ) b ) w } ) )
5		fveq2 6191	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
6	5, 5	jca 554	. . . . 5 |- ( g = G -> ( (Vtx ` g ) = (Vtx ` G ) /\ (Vtx ` g ) = (Vtx ` G ) ) )
7	6	adantl 482	. . . 4 |- ( ( n = N /\ g = G ) -> ( (Vtx ` g ) = (Vtx ` G ) /\ (Vtx ` g ) = (Vtx ` G ) ) )
8	4, 7	el2mpt2cl 7251	. . 3 |- ( A. n e. NN0 A. g e. _V ( (Vtx ` g ) e. _V /\ (Vtx ` g ) e. _V ) -> ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) ) )
9	3, 8	ax-mp 5	. 2 |- ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) )
10		simprl 794	. . 3 |- ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) ) -> ( N e. NN0 /\ G e. _V ) )
11		wwlknon.v	. . . . . . . 8 |- V = (Vtx ` G )
12	11	eleq2i 2693	. . . . . . 7 |- ( A e. V <-> A e. (Vtx ` G ) )
13	11	eleq2i 2693	. . . . . . 7 |- ( B e. V <-> B e. (Vtx ` G ) )
14	12, 13	anbi12i 733	. . . . . 6 |- ( ( A e. V /\ B e. V ) <-> ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) )
15	14	biimpri 218	. . . . 5 |- ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( A e. V /\ B e. V ) )
16	15	adantl 482	. . . 4 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> ( A e. V /\ B e. V ) )
17	16	adantl 482	. . 3 |- ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) ) -> ( A e. V /\ B e. V ) )
18		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
19	18	wspthnon 26743	. . . . . 6 |- ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( W e. ( A ( N WSPathsNOn G ) B ) <-> ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) ) )
20	19	biimpd 219	. . . . 5 |- ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) -> ( W e. ( A ( N WSPathsNOn G ) B ) -> ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) ) )
21	20	adantl 482	. . . 4 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) -> ( W e. ( A ( N WSPathsNOn G ) B ) -> ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) ) )
22	21	impcom 446	. . 3 |- ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) ) -> ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) )
23	10, 17, 22	3jca 1242	. 2 |- ( ( W e. ( A ( N WSPathsNOn G ) B ) /\ ( ( N e. NN0 /\ G e. _V ) /\ ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) ) ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) /\ ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) ) )
24	9, 23	mpdan 702	1 |- ( W e. ( A ( N WSPathsNOn G ) B ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( A e. V /\ B e. V ) /\ ( W e. ( A ( N WWalksNOn G ) B ) /\ E. f f ( A (SPathsOn ` G ) B ) W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   NN0cn0 11292  Vtxcvtx 25874  SPathsOncspthson 26611   WWalksNOn cwwlksnon 26719   WSPathsNOn cwwspthsnon 26721

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-fal 1489  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-wwlksnon 26724  df-wspthsnon 26726

This theorem is referenced by:  wspthneq1eq2  26745  wspthsnonn0vne  26813  wspthsswwlknon  26817