Theorem wspthsnon 26739

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

Hypothesis

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

Assertion

Ref	Expression
wspthsnon	|- ( ( N e. NN0 /\ G e. U ) -> ( N WSPathsNOn G ) = ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a (SPathsOn ` G ) b ) w } ) )

Distinct variable groups:   G, a, b, w   N, a, b, w   V, a, b   f, G, a, b, w   f, N

Allowed substitution hints:   U( w, f, a, b)   V( w, f)

Proof of Theorem wspthsnon

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

Step	Hyp	Ref	Expression
1		df-wspthsnon 26726	. . 3 |- 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 } ) )
2	1	a1i 11	. 2 |- ( ( N e. NN0 /\ G e. U ) -> 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 } ) ) )
3		fveq2 6191	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
4		wwlksnon.v	. . . . . 6 |- V = (Vtx ` G )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
6	5	adantl 482	. . . 4 |- ( ( n = N /\ g = G ) -> (Vtx ` g ) = V )
7		oveq12 6659	. . . . . 6 |- ( ( n = N /\ g = G ) -> ( n WWalksNOn g ) = ( N WWalksNOn G ) )
8	7	oveqd 6667	. . . . 5 |- ( ( n = N /\ g = G ) -> ( a ( n WWalksNOn g ) b ) = ( a ( N WWalksNOn G ) b ) )
9		fveq2 6191	. . . . . . . . 9 |- ( g = G -> (SPathsOn ` g ) = (SPathsOn ` G ) )
10	9	oveqd 6667	. . . . . . . 8 |- ( g = G -> ( a (SPathsOn ` g ) b ) = ( a (SPathsOn ` G ) b ) )
11	10	breqd 4664	. . . . . . 7 |- ( g = G -> ( f ( a (SPathsOn ` g ) b ) w <-> f ( a (SPathsOn ` G ) b ) w ) )
12	11	adantl 482	. . . . . 6 |- ( ( n = N /\ g = G ) -> ( f ( a (SPathsOn ` g ) b ) w <-> f ( a (SPathsOn ` G ) b ) w ) )
13	12	exbidv 1850	. . . . 5 |- ( ( n = N /\ g = G ) -> ( E. f f ( a (SPathsOn ` g ) b ) w <-> E. f f ( a (SPathsOn ` G ) b ) w ) )
14	8, 13	rabeqbidv 3195	. . . 4 |- ( ( n = N /\ g = G ) -> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a (SPathsOn ` g ) b ) w } = { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a (SPathsOn ` G ) b ) w } )
15	6, 6, 14	mpt2eq123dv 6717	. . 3 |- ( ( n = N /\ g = G ) -> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a (SPathsOn ` g ) b ) w } ) = ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a (SPathsOn ` G ) b ) w } ) )
16	15	adantl 482	. 2 |- ( ( ( N e. NN0 /\ G e. U ) /\ ( n = N /\ g = G ) ) -> ( a e. (Vtx ` g ) , b e. (Vtx ` g ) |-> { w e. ( a ( n WWalksNOn g ) b ) | E. f f ( a (SPathsOn ` g ) b ) w } ) = ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a (SPathsOn ` G ) b ) w } ) )
17		simpl 473	. 2 |- ( ( N e. NN0 /\ G e. U ) -> N e. NN0 )
18		elex 3212	. . 3 |- ( G e. U -> G e. _V )
19	18	adantl 482	. 2 |- ( ( N e. NN0 /\ G e. U ) -> G e. _V )
20		fvex 6201	. . . . 5 |- (Vtx ` G ) e. _V
21	4, 20	eqeltri 2697	. . . 4 |- V e. _V
22	21, 21	mpt2ex 7247	. . 3 |- ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a (SPathsOn ` G ) b ) w } ) e. _V
23	22	a1i 11	. 2 |- ( ( N e. NN0 /\ G e. U ) -> ( a e. V , b e. V |-> { w e. ( a ( N WWalksNOn G ) b ) | E. f f ( a (SPathsOn ` G ) b ) w } ) e. _V )
24	2, 16, 17, 19, 23	ovmpt2d 6788	1 |- ( ( N e. NN0 /\ G e. U ) -> ( N WSPathsNOn G ) = ( 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   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   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-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-wspthsnon 26726

This theorem is referenced by:  iswspthsnon  26741