Theorem upgriseupth 27067

Description: The property " <. F , P >. is an Eulerian path on the pseudograph G". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

Hypotheses

Ref	Expression
eupths.i	|- I = (iEdg ` G )
upgriseupth.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
upgriseupth	|- ( G e. UPGraph -> ( F (EulerPaths ` G ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )

Distinct variable groups:   k, F   k, G   k, I   P, k   k, V

Proof of Theorem upgriseupth

Step	Hyp	Ref	Expression
1		eupths.i	. . . 4 |- I = (iEdg ` G )
2	1	iseupthf1o 27062	. . 3 |- ( F (EulerPaths ` G ) P <-> ( F (Walks ` G ) P /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) )
3	2	a1i 11	. 2 |- ( G e. UPGraph -> ( F (EulerPaths ` G ) P <-> ( F (Walks ` G ) P /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) ) )
4		upgriseupth.v	. . . 4 |- V = (Vtx ` G )
5	4, 1	upgriswlk 26537	. . 3 |- ( G e. UPGraph -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
6	5	anbi1d 741	. 2 |- ( G e. UPGraph -> ( ( F (Walks ` G ) P /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) <-> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) ) )
7		simpr 477	. . . . 5 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) -> F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I )
8		simpl2 1065	. . . . 5 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) -> P : ( 0 ... ( # ` F ) ) --> V )
9		simpl3 1066	. . . . 5 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) -> A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
10	7, 8, 9	3jca 1242	. . . 4 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) -> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
11		f1of 6137	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I -> F : ( 0..^ ( # ` F ) ) --> dom I )
12		iswrdi 13309	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) --> dom I -> F e. Word dom I )
13	11, 12	syl 17	. . . . . 6 |- ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I -> F e. Word dom I )
14	13	3anim1i 1248	. . . . 5 |- ( ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
15		simp1 1061	. . . . 5 |- ( ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I )
16	14, 15	jca 554	. . . 4 |- ( ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) -> ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) )
17	10, 16	impbii 199	. . 3 |- ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) <-> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
18	17	a1i 11	. 2 |- ( G e. UPGraph -> ( ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) /\ F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I ) <-> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
19	3, 6, 18	3bitrd 294	1 |- ( G e. UPGraph -> ( F (EulerPaths ` G ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-onto-> dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( I ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 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   dom cdm 5114   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975  Walkscwlks 26492  EulerPathsceupth 27057

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-ifp 1013  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-rmo 2920  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-wlks 26495  df-trls 26589  df-eupth 27058

This theorem is referenced by:  upgreupthi  27068