Theorem wlkvtxiedgg 16341

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
wlkvtxiedgg	|- ( ( G e. W /\ 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   e, W, k

Proof of Theorem wlkvtxiedgg

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

Step	Hyp	Ref	Expression
1		wlkvtxeledg.i	. . 3 |- I = (iEdg ` G )
2	1	wlkvtxeledgg 16339	. 2 |- ( ( G e. W /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
3		eqid 2232	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
4	3	wlkpg 16330	. . . . . . . . . 10 |- ( ( G e. W /\ F (Walks ` G ) P ) -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) )
5	4	adantr 276	. . . . . . . . 9 |- ( ( ( G e. W /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) )
6		elfzofz 10497	. . . . . . . . . 10 |- ( k e. ( 0..^ (♯ ` F ) ) -> k e. ( 0 ... (♯ ` F ) ) )
7	6	adantl 277	. . . . . . . . 9 |- ( ( ( G e. W /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> k e. ( 0 ... (♯ ` F ) ) )
8	5, 7	ffvelcdmd 5813	. . . . . . . 8 |- ( ( ( G e. W /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( P ` k ) e. (Vtx ` G ) )
9		prmg 3814	. . . . . . . 8 |- ( ( P ` k ) e. (Vtx ` G ) -> E. x x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } )
10	8, 9	syl 14	. . . . . . 7 |- ( ( ( G e. W /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> E. x x e. { ( P ` k ) , ( P ` ( k + 1 ) ) } )
11	10	adantr 276	. . . . . 6 |- ( ( ( ( G e. W /\ 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 ) ) } )
12		wlkvg 16323	. . . . . . . . . . 11 |- ( ( G e. W /\ F (Walks ` G ) P ) -> ( F e. _V /\ P e. _V ) )
13	12	simpld 112	. . . . . . . . . 10 |- ( ( G e. W /\ F (Walks ` G ) P ) -> F e. _V )
14		vex 2816	. . . . . . . . . 10 |- k e. _V
15		fvexg 5689	. . . . . . . . . 10 |- ( ( F e. _V /\ k e. _V ) -> ( F ` k ) e. _V )
16	13, 14, 15	sylancl 413	. . . . . . . . 9 |- ( ( G e. W /\ F (Walks ` G ) P ) -> ( F ` k ) e. _V )
17	16	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ( G e. W /\ 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 )
18		simplr 529	. . . . . . . . . 10 |- ( ( ( ( ( G e. W /\ 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 ) ) )
19		simpr 110	. . . . . . . . . 10 |- ( ( ( ( ( G e. W /\ 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 ) ) } )
20	18, 19	sseldd 3239	. . . . . . . . 9 |- ( ( ( ( ( G e. W /\ 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 ) ) )
21		fvmbr 5705	. . . . . . . . 9 |- ( x e. ( I ` ( F ` k ) ) -> ( F ` k ) I ( I ` ( F ` k ) ) )
22	20, 21	syl 14	. . . . . . . 8 |- ( ( ( ( ( G e. W /\ 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 ) ) )
23		breq1 4112	. . . . . . . 8 |- ( y = ( F ` k ) -> ( y I ( I ` ( F ` k ) ) <-> ( F ` k ) I ( I ` ( F ` k ) ) ) )
24	17, 22, 23	elabd 2962	. . . . . . 7 |- ( ( ( ( ( G e. W /\ 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 ) ) )
25		elfvfvex 5704	. . . . . . . 8 |- ( x e. ( I ` ( F ` k ) ) -> ( I ` ( F ` k ) ) e. _V )
26		elrng 4946	. . . . . . . 8 |- ( ( I ` ( F ` k ) ) e. _V -> ( ( I ` ( F ` k ) ) e. ran I <-> E. y y I ( I ` ( F ` k ) ) ) )
27	20, 25, 26	3syl 17	. . . . . . 7 |- ( ( ( ( ( G e. W /\ 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 ) ) ) )
28	24, 27	mpbird 167	. . . . . 6 |- ( ( ( ( ( G e. W /\ 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 )
29	11, 28	exlimddv 1948	. . . . 5 |- ( ( ( ( G e. W /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) e. ran I )
30		sseq2 3262	. . . . . 6 |- ( e = ( I ` ( F ` k ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
31	30	adantl 277	. . . . 5 |- ( ( ( ( ( G e. W /\ 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 ) ) ) )
32		simpr 110	. . . . 5 |- ( ( ( ( G e. W /\ 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 ) ) )
33	29, 31, 32	rspcedvd 2927	. . . 4 |- ( ( ( ( G e. W /\ 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 )
34	33	ex 115	. . 3 |- ( ( ( G e. W /\ 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 ) )
35	34	ralimdva 2609	. 2 |- ( ( G e. W /\ 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 ) )
36	2, 35	mpd 13	1 |- ( ( G e. W /\ 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 2203   A.wral 2520   E.wrex 2521   _Vcvv 2813   C_ wss 3211   {cpr 3690   class class class wbr 4109   ran crn 4750   -->wf 5348   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   + caddc 8130   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Vtxcvtx 16007  iEdgciedg 16008  Walkscwlks 16312

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-map 6884  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-wlks 16313

This theorem is referenced by: (None)