Theorem wlkiswwlks2lem4 26758

Description: Lemma 4 for wlkiswwlks2 26761. (Contributed by Alexander van der Vekens, 20-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
wlkiswwlks2lem4	|- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )

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

Allowed substitution hints:   E( i)   F( x)   G( x)

Proof of Theorem wlkiswwlks2lem4

Step	Hyp	Ref	Expression
1		wlkiswwlks2lem.f	. . . 4 |- F = ( x e. ( 0..^ ( ( # ` P ) - 1 ) ) |-> ( `' E ` { ( P ` x ) , ( P ` ( x + 1 ) ) } ) )
2	1	wlkiswwlks2lem1 26755	. . 3 |- ( ( P e. Word V /\ 1 <_ ( # ` P ) ) -> ( # ` F ) = ( ( # ` P ) - 1 ) )
3	2	3adant1 1079	. 2 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( # ` F ) = ( ( # ` P ) - 1 ) )
4		lencl 13324	. . . . . . . . . 10 |- ( P e. Word V -> ( # ` P ) e. NN0 )
5	4	3ad2ant2 1083	. . . . . . . . 9 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( # ` P ) e. NN0 )
6	1	wlkiswwlks2lem2 26756	. . . . . . . . 9 |- ( ( ( # ` P ) e. NN0 /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( F ` i ) = ( `' E ` { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
7	5, 6	sylan 488	. . . . . . . 8 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( F ` i ) = ( `' E ` { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
8	7	adantr 481	. . . . . . 7 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) /\ { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> ( F ` i ) = ( `' E ` { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
9	8	fveq2d 6195	. . . . . 6 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) /\ { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> ( E ` ( F ` i ) ) = ( E ` ( `' E ` { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
10		wlkiswwlks2lem.e	. . . . . . . . . . 11 |- E = (iEdg ` G )
11	10	uspgrf1oedg 26068	. . . . . . . . . 10 |- ( G e. USPGraph -> E : dom E -1-1-onto-> (Edg ` G ) )
12	10	rneqi 5352	. . . . . . . . . . . 12 |- ran E = ran (iEdg ` G )
13		edgval 25941	. . . . . . . . . . . 12 |- (Edg ` G ) = ran (iEdg ` G )
14	12, 13	eqtr4i 2647	. . . . . . . . . . 11 |- ran E = (Edg ` G )
15		f1oeq3 6129	. . . . . . . . . . 11 |- ( ran E = (Edg ` G ) -> ( E : dom E -1-1-onto-> ran E <-> E : dom E -1-1-onto-> (Edg ` G ) ) )
16	14, 15	ax-mp 5	. . . . . . . . . 10 |- ( E : dom E -1-1-onto-> ran E <-> E : dom E -1-1-onto-> (Edg ` G ) )
17	11, 16	sylibr 224	. . . . . . . . 9 |- ( G e. USPGraph -> E : dom E -1-1-onto-> ran E )
18	17	3ad2ant1 1082	. . . . . . . 8 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> E : dom E -1-1-onto-> ran E )
19	18	adantr 481	. . . . . . 7 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> E : dom E -1-1-onto-> ran E )
20		f1ocnvfv2 6533	. . . . . . 7 |- ( ( E : dom E -1-1-onto-> ran E /\ { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> ( E ` ( `' E ` { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
21	19, 20	sylan 488	. . . . . 6 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) /\ { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> ( E ` ( `' E ` { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
22	9, 21	eqtrd 2656	. . . . 5 |- ( ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) /\ { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) -> ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
23	22	ex 450	. . . 4 |- ( ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) /\ i e. ( 0..^ ( ( # ` P ) - 1 ) ) ) -> ( { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
24	23	ralimdva 2962	. . 3 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
25		oveq2 6658	. . . . 5 |- ( ( # ` F ) = ( ( # ` P ) - 1 ) -> ( 0..^ ( # ` F ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
26	25	raleqdv 3144	. . . 4 |- ( ( # ` F ) = ( ( # ` P ) - 1 ) -> ( A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
27	26	imbi2d 330	. . 3 |- ( ( # ` F ) = ( ( # ` P ) - 1 ) -> ( ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
28	24, 27	syl5ibr 236	. 2 |- ( ( # ` F ) = ( ( # ` P ) - 1 ) -> ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) )
29	3, 28	mpcom 38	1 |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E -> A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )

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   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NN0cn0 11292  ..^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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-hash 13118  df-word 13299  df-edg 25940  df-uspgr 26045

This theorem is referenced by:  wlkiswwlks2lem6  26760