Theorem wlkvtxiedg 16056

Description: The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)

Hypothesis

Ref	Expression
wlkvtxeledg.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
wlkvtxiedg	|- ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )

Distinct variable groups:   k, G   k, F   P, k   e, F   e, G   e, I, k   P, e

Proof of Theorem wlkvtxiedg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkv 16038	. . . 4 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2	1	simp1d 1033	. . 3 |- ( F (Walks ` G ) P -> G e. _V )
3		wlkvtxeledg.i	. . . 4 |- I = (iEdg ` G )
4	3	wlkvtxeledgg 16055	. . 3 |- ( ( G e. _V /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
5	2, 4	mpancom 422	. 2 |- ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
6		eqid 2229	. . . . . . . . . . . 12 |- (Vtx ` G ) = (Vtx ` G )
7	6	wlkpg 16047	. . . . . . . . . . 11 |- ( ( G e. _V /\ F (Walks ` G ) P ) -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) )
8	7	adantr 276	. . . . . . . . . 10 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) )
9		elfzofz 10359	. . . . . . . . . . 11 |- ( k e. ( 0..^ (♯ ` F ) ) -> k e. ( 0 ... (♯ ` F ) ) )
10	9	adantl 277	. . . . . . . . . 10 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> k e. ( 0 ... (♯ ` F ) ) )
11	8, 10	ffvelcdmd 5771	. . . . . . . . 9 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( P ` k ) e. (Vtx ` G ) )
12		prmg 3789	. . . . . . . . 9 |- ( ( P ` k ) e. (Vtx ` G ) -> E. x x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } )
13	11, 12	syl 14	. . . . . . . 8 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> E. x x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } )
14	13	adantr 276	. . . . . . 7 |- ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> E. x x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } )
15	1	simp2d 1034	. . . . . . . . . . . 12 |- ( F (Walks ` G ) P -> F e. _V )
16	15	adantl 277	. . . . . . . . . . 11 |- ( ( G e. _V /\ F (Walks ` G ) P ) -> F e. _V )
17		vex 2802	. . . . . . . . . . 11 |- k e. _V
18		fvexg 5646	. . . . . . . . . . 11 |- ( ( F e. _V /\ k e. _V ) -> ( F ` k ) e. _V )
19	16, 17, 18	sylancl 413	. . . . . . . . . 10 |- ( ( G e. _V /\ F (Walks ` G ) P ) -> ( F ` k ) e. _V )
20	19	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( F ` k ) e. _V )
21		simplr 528	. . . . . . . . . . 11 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
22		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } )
23	21, 22	sseldd 3225	. . . . . . . . . 10 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> x e. ( I ` ( F ` k ) ) )
24		fvmbr 5662	. . . . . . . . . 10 |- ( x e. ( I ` ( F ` k ) ) -> ( F ` k ) I ( I ` ( F ` k ) ) )
25	23, 24	syl 14	. . . . . . . . 9 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( F ` k ) I ( I ` ( F ` k ) ) )
26		breq1 4086	. . . . . . . . 9 |- ( y = ( F ` k ) -> ( y I ( I ` ( F ` k ) ) <-> ( F ` k ) I ( I ` ( F ` k ) ) ) )
27	20, 25, 26	elabd 2948	. . . . . . . 8 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> E. y y I ( I ` ( F ` k ) ) )
28		elfvex 5661	. . . . . . . . 9 |- ( x e. ( I ` ( F ` k ) ) -> ( I ` ( F ` k ) ) e. _V )
29		elrng 4913	. . . . . . . . 9 |- ( ( I ` ( F ` k ) ) e. _V -> ( ( I ` ( F ` k ) ) e. ran I <-> E. y y I ( I ` ( F ` k ) ) ) )
30	23, 28, 29	3syl 17	. . . . . . . 8 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( ( I ` ( F ` k ) ) e. ran I <-> E. y y I ( I ` ( F ` k ) ) ) )
31	27, 30	mpbird 167	. . . . . . 7 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( I ` ( F ` k ) ) e. ran I )
32	14, 31	exlimddv 1945	. . . . . 6 |- ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) e. ran I )
33		sseq2 3248	. . . . . . 7 |- ( e = ( I ` ( F ` k ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
34	33	adantl 277	. . . . . 6 |- ( ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ e = ( I ` ( F ` k ) ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
35		simpr 110	. . . . . 6 |- ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
36	32, 34, 35	rspcedvd 2913	. . . . 5 |- ( ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
37	36	ex 115	. . . 4 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) )
38	37	ralimdva 2597	. . 3 |- ( ( G e. _V /\ F (Walks ` G ) P ) -> ( A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) )
39	2, 38	mpancom 422	. 2 |- ( F (Walks ` G ) P -> ( A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) )
40	5, 39	mpd 13	1 |- ( F (Walks ` G ) P -> A. k e. ( 0..^ (♯ ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2799   C_ wss 3197   {cpr 3667   class class class wbr 4083   ran crn 4720   -->wf 5314   ` cfv 5318  (class class class)co 6001   0cc0 7999   1c1 8000   + caddc 8002   ...cfz 10204  ..^cfzo 10338  ♯chash 10997  Vtxcvtx 15813  iEdgciedg 15814  Walkscwlks 16030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-map 6797  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-wlks 16031

This theorem is referenced by:  wlkvtxedg  16074