Theorem wlkiswwlks2lem5 26759

Description: Lemma 5 for wlkiswwlks2 26761. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

Hypotheses

Ref	Expression
wlkiswwlks2lem.f	|- F = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) )
wlkiswwlks2lem.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
wlkiswwlks2lem5	|- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> F e. Word dom E ) )

Distinct variable groups:   x, P   x, E   x, V   i, F   i, G   P, i   i, V, x   i, E   x, G

Allowed substitution hint:   F( x)

Proof of Theorem wlkiswwlks2lem5

Step	Hyp	Ref	Expression
1		wlkiswwlks2lem.e	. . . . . . . . 9 |- E = (iEdg ` G )
2	1	uspgrf1oedg 26068	. . . . . . . 8 |- ( G e. USPGraph -> E : dom E -1-1-onto-> (Edg ` G ) )
3	1	rneqi 5352	. . . . . . . . . . 11 |- ran E = ran (iEdg ` G )
4		edgval 25941	. . . . . . . . . . 11 |- (Edg ` G ) = ran (iEdg ` G )
5	3, 4	eqtr4i 2647	. . . . . . . . . 10 |- ran E = (Edg ` G )
6	5	a1i 11	. . . . . . . . 9 |- ( G e. USPGraph -> ran E = (Edg ` G ) )
7	6	f1oeq3d 6134	. . . . . . . 8 |- ( G e. USPGraph -> ( E : dom E -1-1-onto-> ran E <-> E : dom E -1-1-onto-> (Edg ` G ) ) )
8	2, 7	mpbird 247	. . . . . . 7 |- ( G e. USPGraph -> E : dom E -1-1-onto-> ran E )
9	8	3ad2ant1 1082	. . . . . 6 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> E : dom E -1-1-onto-> ran E )
10	9	ad2antrr 762	. . . . 5 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) /\ x e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> E : dom E -1-1-onto-> ran E )
11		simpr 477	. . . . . . . 8 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ x e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> x e. ( 0..^ ( ( # ` P ) - 1 ) ) )
12		fveq2 6191	. . . . . . . . . . 11 |- ( i = x -> ( P ` i ) = ( P ` x ) )
13		oveq1 6657	. . . . . . . . . . . 12 |- ( i = x -> ( i + 1 ) = ( x + 1 ) )
14	13	fveq2d 6195	. . . . . . . . . . 11 |- ( i = x -> ( P ` ( i + 1 ) ) = ( P ` ( x + 1 ) ) )
15	12, 14	preq12d 4276	. . . . . . . . . 10 |- ( i = x -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` x ) , ( P ` ( x + 1 ) ) } )
16	15	eleq1d 2686	. . . . . . . . 9 |- ( i = x -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E <-> { ( P ` x ) , ( P ` ( x + 1 ) ) } e. ran E ) )
17	16	adantl 482	. . . . . . . 8 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ x e. ( 0..^ ( ( # ` P ) - 1 ) ) ) /\ i = x ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E <-> { ( P ` x ) , ( P ` ( x + 1 ) ) } e. ran E ) )
18	11, 17	rspcdv 3312	. . . . . . 7 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ x e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> { ( P ` x ) , ( P ` ( x + 1 ) ) } e. ran E ) )
19	18	impancom 456	. . . . . 6 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } e. ran E ) )
20	19	imp 445	. . . . 5 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) /\ x e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } e. ran E )
21		f1ocnvdm 6540	. . . . 5 |- ( ( E : dom E -1-1-onto-> ran E /\ { ( P ` x ) , ( P ` ( x + 1 ) ) } e. ran E ) -> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) e. dom E )
22	10, 20, 21	syl2anc 693	. . . 4 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) /\ x e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) e. dom E )
23		wlkiswwlks2lem.f	. . . 4 |- F = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) )
24	22, 23	fmptd 6385	. . 3 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> F : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom E )
25		iswrdi 13309	. . 3 |- ( F : ( 0..^ ( ( # ` P ) - 1 ) ) --> dom E -> F e. Word dom E )
26	24, 25	syl 17	. 2 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> F e. Word dom E )
27	26	ex 450	1 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> F e. Word dom E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266  ..^cfzo 12465   #chash 13117  Word cword 13291  iEdgciedg 25875  Edgcedg 25939   USPGraph cuspgr 26043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-word 13299  df-edg 25940  df-uspgr 26045

This theorem is referenced by:  wlkiswwlks2lem6  26760