Theorem clwwlknon2x 16285

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 16284	. 2 |- ( X C 2 ) = { w e. ( 2 ClWWalksN G ) | ( w ` 0 ) = X }
3		clwwlkn2 16271	. . . . 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 1016	. . . . . 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 2235	. . . . . . . . 9 |- (Vtx ` G ) = V
10	9	wrdeqi 11135	. . . . . . . 8 |- Word (Vtx ` G ) = Word V
11	10	eleq2i 2298	. . . . . . 7 |- ( w e. Word (Vtx ` G ) <-> w e. Word V )
12		df-3an 1006	. . . . . . . 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 2298	. . . . . . . . . 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 2787	. 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 2252	1 |- ( X C 2 ) = { w e. Word V | ( (♯ ` w ) = 2 /\ { ( w ` 0 ) , ( w ` 1 ) } e. E /\ ( w ` 0 ) = X ) }

Syntax hints:   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   {crab 2514   {cpr 3670   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   2c2 9193  ♯chash 11036  Word cword 11112  Vtxcvtx 15862  Edgcedg 15907   ClWWalksN cclwwlkn 16253  ClWWalksNOncclwwlknon 16276

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-lsw 11158  df-ndx 13084  df-slot 13085  df-base 13087  df-vtx 15864  df-clwwlk 16242  df-clwwlkn 16254  df-clwwlknon 16277

This theorem is referenced by:  s2elclwwlknon2  16286