Theorem usgra2adedgwlkon 30333

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 23292	. . . . . . 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 23318	. . . . . . . . . 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 23318	. . . . . . . . 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 1168	. . . . . 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 30329	. . . . 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 5722	. . . . . . 7 |- ( `' E ` { A , B } ) e. _V
16		fvex 5722	. . . . . . 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 23519	. . . . 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 23475	. . . . . . . 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 23471	. . . . . . . . . 10 |- ( # ` F ) = 2
28	27	fveq2i 5715	. . . . . . . . 9 |- ( P ` ( # ` F ) ) = ( P ` 2 )
29	19	2wlklemC 23477	. . . . . . . . 9 |- ( C e. V -> ( P ` 2 ) = C )
30	28, 29	syl5eq 2487	. . . . . . . 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 969	. . . 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 4555	. . . . . . 7 |- { <. 0 , ( `' E ` { A , B } ) >. , <. 1 , ( `' E ` { B , C } ) >. } e. _V
38	18, 37	eqeltri 2513	. . . . . 6 |- F e. _V
39		tpex 6400	. . . . . . 7 |- { <. 0 , A >. , <. 1 , B >. , <. 2 , C >. } e. _V
40	19, 39	eqeltri 2513	. . . . . 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 833	. . . 4 |- ( ( V USGrph E /\ ( { A , B } e. ran E /\ { B , C } e. ran E ) ) -> ( A e. V /\ C e. V ) )
46		iswlkon 23452	. . . 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 1218	. . 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 965   = wceq 1369   e. wcel 1756   _Vcvv 2993   {cpr 3900   {ctp 3902   <.cop 3904   class class class wbr 4313   `'ccnv 4860   ran crn 4862   ` cfv 5439  (class class class)co 6112   0cc0 9303   1c1 9304   2c2 10392   #chash 12124   USGrph cusg 23286   Walks cwalk 23427   WalkOn cwlkon 23431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-usgra 23288  df-wlk 23437  df-wlkon 23443

This theorem is referenced by:  usg2wlkon  30336