Theorem wlkvtxiedg 16215

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 16196	. . . 4 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2	1	simp1d 1035	. . 3 |- ( F (Walks ` G ) P -> G e. _V )
3		wlkvtxeledg.i	. . . 4 |- I = (iEdg ` G )
4	3	wlkvtxeledgg 16214	. . 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 2231	. . . . . . . . . . . 12 |- (Vtx ` G ) = (Vtx ` G )
7	6	wlkpg 16205	. . . . . . . . . . 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 10398	. . . . . . . . . . 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 5783	. . . . . . . . 9 |- ( ( ( G e. _V /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( P ` k ) e. (Vtx ` G ) )
12		prmg 3794	. . . . . . . . 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 1036	. . . . . . . . . . . 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 2805	. . . . . . . . . . 11 |- k e. _V
18		fvexg 5658	. . . . . . . . . . 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 3228	. . . . . . . . . 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 5674	. . . . . . . . . 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 4091	. . . . . . . . 9 |- ( y = ( F ` k ) -> ( y I ( I ` ( F ` k ) ) <-> ( F ` k ) I ( I ` ( F ` k ) ) ) )
27	20, 25, 26	elabd 2951	. . . . . . . 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 5673	. . . . . . . . 9 |- ( x e. ( I ` ( F ` k ) ) -> ( I ` ( F ` k ) ) e. _V )
29		elrng 4921	. . . . . . . . 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 1947	. . . . . 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 3251	. . . . . . 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 2916	. . . . 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 2599	. . 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 1397   E.wex 1540   e. wcel 2202   A.wral 2510   E.wrex 2511   _Vcvv 2802   C_ wss 3200   {cpr 3670   class class class wbr 4088   ran crn 4726   -->wf 5322   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   + caddc 8035   ...cfz 10243  ..^cfzo 10377  ♯chash 11038  Vtxcvtx 15882  iEdgciedg 15883  Walkscwlks 16187

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-fz 10244  df-fzo 10378  df-ihash 11039  df-word 11118  df-ndx 13103  df-slot 13104  df-base 13106  df-edgf 15875  df-vtx 15884  df-iedg 15885  df-wlks 16188

This theorem is referenced by:  wlkvtxedg  16233