Theorem usgra2adedgwlkon 24742

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 24465	. . . . . . 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 24504	. . . . . . . . . 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 24504	. . . . . . . . 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 1176	. . . . . 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 24738	. . . . 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 5882	. . . . . . 7 |- ( `' E ` { A , B } ) e. _V
16		fvex 5882	. . . . . . 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 24727	. . . . 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 24683	. . . . . . . 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 24679	. . . . . . . . . 10 |- ( # ` F ) = 2
28	27	fveq2i 5875	. . . . . . . . 9 |- ( P ` ( # ` F ) ) = ( P ` 2 )
29	19	2wlklemC 24685	. . . . . . . . 9 |- ( C e. V -> ( P ` 2 ) = C )
30	28, 29	syl5eq 2510	. . . . . . . 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 977	. . . 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 4698	. . . . . . 7 |- { <. 0 , ( `' E ` { A , B } ) >. , <. 1 , ( `' E ` { B , C } ) >. } e. _V
38	18, 37	eqeltri 2541	. . . . . 6 |- F e. _V
39		tpex 6598	. . . . . . 7 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
40	19, 39	eqeltri 2541	. . . . . 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 837	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ C e. V ) )
46		iswlkon 24661	. . . 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 1228	. . 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 973   = wceq 1395   e. wcel 1819   _Vcvv 3109   {cpr 4034   {ctp 4036   <.cop 4038   class class class wbr 4456   `'ccnv 5007   ran crn 5009   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   2c2 10606   #chash 12408   USGrph cusg 24457   Walks cwalk 24625   WalkOn cwlkon 24629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-usgra 24460  df-wlk 24635  df-wlkon 24641

This theorem is referenced by:  usg2wlkon  24745