Theorem wlkvtxiedg 16466

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 16447	. . . 4 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2	1	simp1d 1036	. . 3 |- ( F (Walks ` G ) P -> G e. _V )
3		wlkvtxeledg.i	. . . 4 |- I = (iEdg ` G )
4	3	wlkvtxeledgg 16465	. . 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 2234	. . . . . . . . . . . 12 |- (Vtx ` G ) = (Vtx ` G )
7	6	wlkpg 16456	. . . . . . . . . . 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 10519	. . . . . . . . . . 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 5818	. . . . . . . . 9 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( P ` k ) e. (Vtx ` G ) )
12		prmg 3819	. . . . . . . . 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 1037	. . . . . . . . . . . 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 2818	. . . . . . . . . . 11 |- k e. _V
18		fvexg 5694	. . . . . . . . . . 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 529	. . . . . . . . . . 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 3243	. . . . . . . . . 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 5710	. . . . . . . . . 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 4117	. . . . . . . . 9 |- ( y = ( F ` k ) -> ( y I ( I ` ( F ` k ) ) <-> ( F ` k ) I ( I ` ( F ` k ) ) ) )
27	20, 25, 26	elabd 2965	. . . . . . . 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		elfvfvex 5709	. . . . . . . . 9 |- ( x e. ( I ` ( F ` k ) ) -> ( I ` ( F ` k ) ) e. _V )
29		elrng 4951	. . . . . . . . 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 1950	. . . . . 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 3266	. . . . . . 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 2929	. . . . 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 2611	. . 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 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   _Vcvv 2815   C_ wss 3214   {cpr 3695   class class class wbr 4114   ran crn 4755   -->wf 5353   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   ...cfz 10361  ..^cfzo 10498  ♯chash 11163  Vtxcvtx 16133  iEdgciedg 16134  Walkscwlks 16438

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-wlks 16439

This theorem is referenced by:  wlkvtxedg  16484