Theorem usgra2adedgwlkon 24277

Description: In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)

Hypotheses

Ref	Expression
usgra2adedgspth.f	|- F = { <. 0 , ( `' E ` { A , B } ) >. , <. 1 , ( `' E ` { B , C } ) >. }
usgra2adedgspth.p	|- P = { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. }

Assertion

Ref	Expression
usgra2adedgwlkon	|- ( V USGrph E -> ( ( { A , B } e. ran E /\ { B , C } e. ran E ) -> F ( A ( V WalkOn E ) C ) P ) )

Proof of Theorem usgra2adedgwlkon

Step	Hyp	Ref	Expression
1		usgrav 24001	. . . . . . 7 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
2	1	adantr 465	. . . . . 6 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( V e. _V /\ E e. _V ) )
3		usgraedgrnv 24039	. . . . . . . . . 10 |- ( ( V USGrph E /\ { A , B } e. ran E ) -> ( A e. V /\ B e. V ) )
4	3	ancomd 451	. . . . . . . . 9 |- ( ( V USGrph E /\ { A , B } e. ran E ) -> ( B e. V /\ A e. V ) )
5	4	adantrr 716	. . . . . . . 8 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( B e. V /\ A e. V ) )
6	5	simprd 463	. . . . . . 7 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> A e. V )
7	3	adantrr 716	. . . . . . . 8 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ B e. V ) )
8	7	simprd 463	. . . . . . 7 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> B e. V )
9		usgraedgrnv 24039	. . . . . . . . 9 |- ( ( V USGrph E /\ { B , C } e. ran E ) -> ( B e. V /\ C e. V ) )
10	9	adantrl 715	. . . . . . . 8 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( B e. V /\ C e. V ) )
11	10	simprd 463	. . . . . . 7 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> C e. V )
12	6, 8, 11	3jca 1171	. . . . . 6 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ B e. V /\ C e. V ) )
13	2, 12	jca 532	. . . . 5 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ B e. V /\ C e. V ) ) )
14		usgra2adedgspthlem1 24273	. . . . 5 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( ( E ` ( `' E ` { A , B } ) ) = { A , B } /\ ( E ` ( `' E ` { B , C } ) ) = { B , C } ) )
15		fvex 5867	. . . . . . 7 |- ( `' E ` { A , B } ) e. _V
16		fvex 5867	. . . . . . 7 |- ( `' E ` { B , C } ) e. _V
17	15, 16	pm3.2i 455	. . . . . 6 |- ( ( `' E ` { A , B } ) e. _V /\ ( `' E ` { B , C } ) e. _V )
18		usgra2adedgspth.f	. . . . . 6 |- F = { <. 0 , ( `' E ` { A , B } ) >. , <. 1 , ( `' E ` { B , C } ) >. }
19		usgra2adedgspth.p	. . . . . 6 |- P = { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. }
20	17, 18, 19	constr2wlk 24262	. . . . 5 |- ( ( ( V e. _V /\ E e. _V ) /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( E ` ( `' E ` { A , B } ) ) = { A , B } /\ ( E ` ( `' E ` { B , C } ) ) = { B , C } ) -> F ( V Walks E ) P ) )
21	13, 14, 20	sylc 60	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> F ( V Walks E ) P )
22	3	ex 434	. . . . . . 7 |- ( V USGrph E -> ( { A , B } e. ran E -> ( A e. V /\ B e. V ) ) )
23	9	ex 434	. . . . . . 7 |- ( V USGrph E -> ( { B , C } e. ran E -> ( B e. V /\ C e. V ) ) )
24	22, 23	anim12d 563	. . . . . 6 |- ( V USGrph E -> ( ( { A , B } e. ran E /\ { B , C } e. ran E ) -> ( ( A e. V /\ B e. V ) /\ ( B e. V /\ C e. V ) ) ) )
25	19	2wlklemA 24218	. . . . . . . 8 |- ( A e. V -> ( P ` 0 ) = A )
26	25	adantr 465	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> ( P ` 0 ) = A )
27	17, 18	2trllemA 24214	. . . . . . . . . 10 |- ( # ` F ) = 2
28	27	fveq2i 5860	. . . . . . . . 9 |- ( P ` ( # ` F ) ) = ( P ` 2 )
29	19	2wlklemC 24220	. . . . . . . . 9 |- ( C e. V -> ( P ` 2 ) = C )
30	28, 29	syl5eq 2513	. . . . . . . 8 |- ( C e. V -> ( P ` ( # ` F ) ) = C )
31	30	adantl 466	. . . . . . 7 |- ( ( B e. V /\ C e. V ) -> ( P ` ( # ` F ) ) = C )
32	26, 31	anim12i 566	. . . . . 6 |- ( ( ( A e. V /\ B e. V ) /\ ( B e. V /\ C e. V ) ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) )
33	24, 32	syl6 33	. . . . 5 |- ( V USGrph E -> ( ( { A , B } e. ran E /\ { B , C } e. ran E ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
34	33	imp 429	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) )
35		3anass 972	. . . 4 |- ( ( F ( V Walks E ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) <-> ( F ( V Walks E ) P /\ ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
36	21, 34, 35	sylanbrc 664	. . 3 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( F ( V Walks E ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) )
37		prex 4682	. . . . . . 7 |- { <. 0 , ( `' E ` { A , B } ) >. , <. 1 , ( `' E ` { B , C } ) >. } e. _V
38	18, 37	eqeltri 2544	. . . . . 6 |- F e. _V
39		tpex 6574	. . . . . . 7 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
40	19, 39	eqeltri 2544	. . . . . 6 |- P e. _V
41	38, 40	pm3.2i 455	. . . . 5 |- ( F e. _V /\ P e. _V )
42	41	a1i 11	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( F e. _V /\ P e. _V ) )
43	3	simpld 459	. . . . 5 |- ( ( V USGrph E /\ { A , B } e. ran E ) -> A e. V )
44	9	simprd 463	. . . . 5 |- ( ( V USGrph E /\ { B , C } e. ran E ) -> C e. V )
45	43, 44	anim12dan 834	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ C e. V ) )
46		iswlkon 24196	. . . 4 |- ( ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) /\ ( A e. V /\ C e. V ) ) -> ( F ( A ( V WalkOn E ) C ) P <-> ( F ( V Walks E ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
47	2, 42, 45, 46	syl3anc 1223	. . 3 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( F ( A ( V WalkOn E ) C ) P <-> ( F ( V Walks E ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
48	36, 47	mpbird 232	. 2 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> F ( A ( V WalkOn E ) C ) P )
49	48	ex 434	1 |- ( V USGrph E -> ( ( { A , B } e. ran E /\ { B , C } e. ran E ) -> F ( A ( V WalkOn E ) C ) P ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   _Vcvv 3106   {cpr 4022   {ctp 4024   <.cop 4026   class class class wbr 4440   `'ccnv 4991   ran crn 4993   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482   2c2 10574   #chash 12360   USGrph cusg 23993   Walks cwalk 24160   WalkOn cwlkon 24164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-usgra 23996  df-wlk 24170  df-wlkon 24176

This theorem is referenced by:  usg2wlkon  24280