Theorem numclwwlkovf2 27217

Description: Value of operation F for argument 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 28-May-2021.)

Hypotheses

Ref	Expression
numclwwlkovf.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
numclwwlkffin.v	|- V = (Vtx ` G )
numclwwlkovfel2.e	|- E = (Edg ` G )

Assertion

Ref	Expression
numclwwlkovf2	|- ( ( G e. USGraph /\ X e. V ) -> ( X F 2 ) = { w e. Word V | ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) } )

Distinct variable groups:   n, G, v, w   n, V, v   n, X, v, w   w, V

Allowed substitution hints:   E( w, v, n)   F( w, v, n)

Proof of Theorem numclwwlkovf2

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( G e. USGraph /\ X e. V ) -> X e. V )
2		2nn 11185	. . 3 |- 2 e. NN
3		numclwwlkovf.f	. . . 4 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
4	3	numclwwlkovf 27213	. . 3 |- ( ( X e. V /\ 2 e. NN ) -> ( X F 2 ) = { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X } )
5	1, 2, 4	sylancl 694	. 2 |- ( ( G e. USGraph /\ X e. V ) -> ( X F 2 ) = { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X } )
6		clwwlksn2 26910	. . . . . 6 |- ( w e. ( 2 ClWWalksN G ) <-> ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
7	6	anbi1i 731	. . . . 5 |- ( ( w e. ( 2 ClWWalksN G ) /\ ( w ` 0 ) = X ) <-> ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) )
8	7	a1i 11	. . . 4 |- ( ( G e. USGraph /\ X e. V ) -> ( ( w e. ( 2 ClWWalksN G ) /\ ( w ` 0 ) = X ) <-> ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) ) )
9		anass 681	. . . . 5 |- ( ( ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) ) /\ ( w ` 0 ) = X ) <-> ( w e. Word V /\ ( ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) /\ ( w ` 0 ) = X ) ) )
10		df-3an 1039	. . . . . . . 8 |- ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) <-> ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
11		ancom 466	. . . . . . . . . 10 |- ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) ) <-> ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 2 ) )
12		numclwwlkffin.v	. . . . . . . . . . . . . 14 |- V = (Vtx ` G )
13	12	eqcomi 2631	. . . . . . . . . . . . 13 |- (Vtx ` G ) = V
14	13	wrdeqi 13328	. . . . . . . . . . . 12 |- Word (Vtx ` G ) = Word V
15	14	eleq2i 2693	. . . . . . . . . . 11 |- ( w e. Word (Vtx ` G ) <-> w e. Word V )
16	15	anbi1i 731	. . . . . . . . . 10 |- ( ( w e. Word (Vtx ` G ) /\ ( # ` w ) = 2 ) <-> ( w e. Word V /\ ( # ` w ) = 2 ) )
17	11, 16	bitri 264	. . . . . . . . 9 |- ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) ) <-> ( w e. Word V /\ ( # ` w ) = 2 ) )
18		numclwwlkovfel2.e	. . . . . . . . . . 11 |- E = (Edg ` G )
19	18	eqcomi 2631	. . . . . . . . . 10 |- (Edg ` G ) = E
20	19	eleq2i 2693	. . . . . . . . 9 |- ( { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) <-> { ( w ` 0 ) , ( w ` 1 ) } e. E )
21	17, 20	anbi12i 733	. . . . . . . 8 |- ( ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) <-> ( ( w e. Word V /\ ( # ` w ) = 2 ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) )
22	10, 21	bitri 264	. . . . . . 7 |- ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) <-> ( ( w e. Word V /\ ( # ` w ) = 2 ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) )
23		anass 681	. . . . . . 7 |- ( ( ( w e. Word V /\ ( # ` w ) = 2 ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) ) )
24	22, 23	bitri 264	. . . . . 6 |- ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) ) )
25	24	anbi1i 731	. . . . 5 |- ( ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) <-> ( ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) ) /\ ( w ` 0 ) = X ) )
26		df-3an 1039	. . . . . 6 |- ( ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) <-> ( ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) /\ ( w ` 0 ) = X ) )
27	26	anbi2i 730	. . . . 5 |- ( ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) ) <-> ( w e. Word V /\ ( ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) /\ ( w ` 0 ) = X ) ) )
28	9, 25, 27	3bitr4i 292	. . . 4 |- ( ( ( ( # ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) ) )
29	8, 28	syl6bb 276	. . 3 |- ( ( G e. USGraph /\ X e. V ) -> ( ( w e. ( 2 ClWWalksN G ) /\ ( w ` 0 ) = X ) <-> ( w e. Word V /\ ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) ) ) )
30	29	rabbidva2 3186	. 2 |- ( ( G e. USGraph /\ X e. V ) -> { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X } = { w e. Word V | ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) } )
31	5, 30	eqtrd 2656	1 |- ( ( G e. USGraph /\ X e. V ) -> ( X F 2 ) = { w e. Word V | ( ( # ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   {cpr 4179   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   NNcn 11020   2c2 11070   #chash 13117  Word cword 13291  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   ClWWalksN cclwwlksn 26876

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-map 7859  df-pm 7860  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwwlkovf2num  27218