Theorem clwwlknon2x 16556

Description: The set of closed walks on vertex X of length 2 in a graph G as words over the set of vertices, definition of ClWWalksN expanded. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Mar-2022.)

Hypotheses

Ref	Expression
clwwlknon2.c	|- C = (ClWWalksNOn ` G )
clwwlknon2x.v	|- V = (Vtx ` G )
clwwlknon2x.e	|- E = (Edg ` G )

Assertion

Ref	Expression
clwwlknon2x	|- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }

Distinct variable groups:   w, G   w, X

Allowed substitution hints:   C( w)   E( w)   V( w)

Proof of Theorem clwwlknon2x

Step	Hyp	Ref	Expression
1		clwwlknon2.c	. . 3 |- C = (ClWWalksNOn ` G )
2	1	clwwlknon2 16555	. 2 |- ( X C 2 ) = { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X }
3		clwwlkn2 16542	. . . . 5 |- ( w e. ( 2 ClWWalksN G ) <-> ( (♯ ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
4	3	anbi1i 458	. . . 4 |- ( ( 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 ) )
5		3anan12 1017	. . . . . 6 |- ( ( (♯ ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) <-> ( w e. Word (Vtx ` G ) /\ ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) )
6	5	anbi1i 458	. . . . 5 |- ( ( ( (♯ ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) <-> ( ( w e. Word (Vtx ` G ) /\ ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) /\ ( w ` 0 ) = X ) )
7		anass 401	. . . . . 6 |- ( ( ( w e. Word (Vtx ` G ) /\ ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) ) /\ ( w ` 0 ) = X ) <-> ( w e. Word (Vtx ` G ) /\ ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) ) )
8		clwwlknon2x.v	. . . . . . . . . 10 |- V = (Vtx ` G )
9	8	eqcomi 2238	. . . . . . . . 9 |- (Vtx ` G ) = V
10	9	wrdeqi 11272	. . . . . . . 8 |- Word (Vtx ` G ) = Word V
11	10	eleq2i 2301	. . . . . . 7 |- ( w e. Word (Vtx ` G ) <-> w e. Word V )
12		df-3an 1007	. . . . . . . 8 |- ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) <-> ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) /\ ( w ` 0 ) = X ) )
13		clwwlknon2x.e	. . . . . . . . . . 11 |- E = (Edg ` G )
14	13	eleq2i 2301	. . . . . . . . . 10 |- ( { ( w ` 0 ) , ( w ` 1 ) } e. E <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) )
15	14	anbi2i 457	. . . . . . . . 9 |- ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) <-> ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
16	15	anbi1i 458	. . . . . . . 8 |- ( ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) /\ ( w ` 0 ) = X ) <-> ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) )
17	12, 16	bitr2i 185	. . . . . . 7 |- ( ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) /\ ( w ` 0 ) = X ) <-> ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) )
18	11, 17	anbi12i 460	. . . . . 6 |- ( ( w e. Word (Vtx ` G ) /\ ( ( (♯ ` w ) = 2 /\ { ( 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 ) ) )
19	7, 18	bitri 184	. . . . 5 |- ( ( ( w e. Word (Vtx ` G ) /\ ( (♯ ` w ) = 2 /\ { ( 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 ) ) )
20	6, 19	bitri 184	. . . 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 ) ) )
21	4, 20	bitri 184	. . 3 |- ( ( w e. ( 2 ClWWalksN G ) /\ ( w ` 0 ) = X ) <-> ( w e. Word V /\ ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) ) )
22	21	rabbia2 2800	. 2 |- { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X } = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }
23	2, 22	eqtri 2255	1 |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }

Syntax hints:   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   {cpr 3695   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   2c2 9305  ♯chash 11163  Word cword 11249  Vtxcvtx 16133  Edgcedg 16178   ClWWalksN cclwwlkn 16524  ClWWalksNOncclwwlknon 16547

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlk 16513  df-clwwlkn 16525  df-clwwlknon 16548

This theorem is referenced by:  s2elclwwlknon2  16557