Theorem usgra2adedgwlkon 25343

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 25065	. . . . . . 7 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
2	1	adantr 467	. . . . . 6 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( V e. _V /\ E e. _V ) )
3		usgraedgrnv 25104	. . . . . . . . . 10 |- ( ( V USGrph E /\ { A , B } e. ran E ) -> ( A e. V /\ B e. V ) )
4	3	ancomd 453	. . . . . . . . 9 |- ( ( V USGrph E /\ { A , B } e. ran E ) -> ( B e. V /\ A e. V ) )
5	4	adantrr 723	. . . . . . . 8 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( B e. V /\ A e. V ) )
6	5	simprd 465	. . . . . . 7 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> A e. V )
7	3	adantrr 723	. . . . . . . 8 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ B e. V ) )
8	7	simprd 465	. . . . . . 7 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> B e. V )
9		usgraedgrnv 25104	. . . . . . . . 9 |- ( ( V USGrph E /\ { B , C } e. ran E ) -> ( B e. V /\ C e. V ) )
10	9	adantrl 722	. . . . . . . 8 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( B e. V /\ C e. V ) )
11	10	simprd 465	. . . . . . 7 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> C e. V )
12	6, 8, 11	3jca 1188	. . . . . 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 535	. . . . 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 25339	. . . . 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 5875	. . . . . . 7 |- ( `' E ` { A , B } ) e. _V
16		fvex 5875	. . . . . . 7 |- ( `' E ` { B , C } ) e. _V
17	15, 16	pm3.2i 457	. . . . . 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 25328	. . . . 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 62	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> F ( V Walks E ) P )
22	3	ex 436	. . . . . . 7 |- ( V USGrph E -> ( { A , B } e. ran E -> ( A e. V /\ B e. V ) ) )
23	9	ex 436	. . . . . . 7 |- ( V USGrph E -> ( { B , C } e. ran E -> ( B e. V /\ C e. V ) ) )
24	22, 23	anim12d 566	. . . . . 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 25284	. . . . . . . 8 |- ( A e. V -> ( P ` 0 ) = A )
26	25	adantr 467	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> ( P ` 0 ) = A )
27	17, 18	2trllemA 25280	. . . . . . . . . 10 |- ( # ` F ) = 2
28	27	fveq2i 5868	. . . . . . . . 9 |- ( P ` ( # ` F ) ) = ( P ` 2 )
29	19	2wlklemC 25286	. . . . . . . . 9 |- ( C e. V -> ( P ` 2 ) = C )
30	28, 29	syl5eq 2497	. . . . . . . 8 |- ( C e. V -> ( P ` ( # ` F ) ) = C )
31	30	adantl 468	. . . . . . 7 |- ( ( B e. V /\ C e. V ) -> ( P ` ( # ` F ) ) = C )
32	26, 31	anim12i 570	. . . . . 6 |- ( ( ( A e. V /\ B e. V ) /\ ( B e. V /\ C e. V ) ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) )
33	24, 32	syl6 34	. . . . 5 |- ( V USGrph E -> ( ( { A , B } e. ran E /\ { B , C } e. ran E ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) ) )
34	33	imp 431	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = C ) )
35		3anass 989	. . . 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 670	. . 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 4642	. . . . . . 7 |- { <. 0 , ( `' E ` { A , B } ) >. , <. 1 , ( `' E ` { B , C } ) >. } e. _V
38	18, 37	eqeltri 2525	. . . . . 6 |- F e. _V
39		tpex 6590	. . . . . . 7 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
40	19, 39	eqeltri 2525	. . . . . 6 |- P e. _V
41	38, 40	pm3.2i 457	. . . . 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 461	. . . . 5 |- ( ( V USGrph E /\ { A , B } e. ran E ) -> A e. V )
44	9	simprd 465	. . . . 5 |- ( ( V USGrph E /\ { B , C } e. ran E ) -> C e. V )
45	43, 44	anim12dan 848	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ C e. V ) )
46		iswlkon 25262	. . . 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 1268	. . 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 236	. 2 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> F ( A ( V WalkOn E ) C ) P )
49	48	ex 436	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   _Vcvv 3045   {cpr 3970   {ctp 3972   <.cop 3974   class class class wbr 4402   `'ccnv 4833   ran crn 4835   ` cfv 5582  (class class class)co 6290   0cc0 9539   1c1 9540   2c2 10659   #chash 12515   USGrph cusg 25057   Walks cwalk 25226   WalkOn cwlkon 25230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-usgra 25060  df-wlk 25236  df-wlkon 25242

This theorem is referenced by:  usg2wlkon  25346