Theorem wlkvtxiedgg 16143

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 16141	. 2 |- ( ( G e. W /\ F (Walks ` G ) P ) -> A. k e. ( 0..^ (♯ ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
3		eqid 2229	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
4	3	wlkpg 16132	. . . . . . . . . 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 10388	. . . . . . . . . 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 5779	. . . . . . . 8 |- ( ( ( G e. W /\ F (Walks ` G ) P ) /\ k e. ( 0..^ (♯ ` F ) ) ) -> ( P ` k ) e. (Vtx ` G ) )
9		prmg 3792	. . . . . . . 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 16125	. . . . . . . . . . 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 2803	. . . . . . . . . 10 |- k e. _V
15		fvexg 5654	. . . . . . . . . 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 528	. . . . . . . . . 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 3226	. . . . . . . . 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 5670	. . . . . . . . 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 4089	. . . . . . . 8 |- ( y = ( F ` k ) -> ( y I ( I ` ( F ` k ) ) <-> ( F ` k ) I ( I ` ( F ` k ) ) ) )
24	17, 22, 23	elabd 2949	. . . . . . 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 5669	. . . . . . . 8 |- ( x e. ( I ` ( F ` k ) ) -> ( I ` ( F ` k ) ) e. _V )
26		elrng 4919	. . . . . . . 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 1945	. . . . 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 3249	. . . . . 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 2914	. . . 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 2597	. 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 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2800   C_ wss 3198   {cpr 3668   class class class wbr 4086   ran crn 4724   -->wf 5320   ` cfv 5324  (class class class)co 6013   0cc0 8022   1c1 8023   + caddc 8025   ...cfz 10233  ..^cfzo 10367  ♯chash 11027  Vtxcvtx 15853  iEdgciedg 15854  Walkscwlks 16114

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-map 6814  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-z 9470  df-dec 9602  df-uz 9746  df-fz 10234  df-fzo 10368  df-ihash 11028  df-word 11104  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856  df-wlks 16115

This theorem is referenced by: (None)