Theorem clwwlknon2x 16676

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 16675	. 2 |- ( X C 2 ) = { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X }
3		clwwlkn2 16662	. . . . 5 |- ( w e. ( 2 ClWWalksN G ) <-> ( (♯ ` w ) = 2 /\ w e. Word (Vtx ` G ) /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
4	3	anbi1i 462	. . . 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 1021	. . . . . 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 462	. . . . 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 405	. . . . . 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 2242	. . . . . . . . 9 |- (Vtx ` G ) = V
10	9	wrdeqi 11327	. . . . . . . 8 |- Word (Vtx ` G ) = Word V
11	10	eleq2i 2305	. . . . . . 7 |- ( w e. Word (Vtx ` G ) <-> w e. Word V )
12		df-3an 1011	. . . . . . . 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 2305	. . . . . . . . . 10 |- ( { ( w ` 0 ) , ( w ` 1 ) } e. E <-> { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) )
15	14	anbi2i 461	. . . . . . . . 9 |- ( ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E ) <-> ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. (Edg ` G ) ) )
16	15	anbi1i 462	. . . . . . . 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 464	. . . . . 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 2806	. 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 2259	1 |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }

This proof depends on syntax axioms:   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   {cpr 3710   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   2c2 9355  ♯chash 11214  Word cword 11304  Vtxcvtx 16253  Edgcedg 16298   ClWWalksN cclwwlkn 16644  ClWWalksNOncclwwlknon 16667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-lsw 11350  df-ndx 13355  df-slot 13356  df-base 13358  df-vtx 16255  df-clwwlk 16633  df-clwwlkn 16645  df-clwwlknon 16668

This theorem is used by:  s2elclwwlknon2  16677