Theorem clwwlknon2x 30104

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	⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 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 30103	. 2 ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
3		clwwlkn2 30045	. . . . 5 ⊢ (𝑤 ∈ (2 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
4	3	anbi1i 624	. . . 4 ⊢ ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
5		3anan12 1095	. . . . . 6 ⊢ (((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))))
6	5	anbi1i 624	. . . . 5 ⊢ ((((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) ∧ (𝑤‘0) = 𝑋))
7		anass 468	. . . . . 6 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)))
8		clwwlknon2x.v	. . . . . . . . . 10 ⊢ 𝑉 = (Vtx‘𝐺)
9	8	eqcomi 2742	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 14451	. . . . . . . 8 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2825	. . . . . . 7 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12		df-3an 1088	. . . . . . . 8 ⊢ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋))
13		clwwlknon2x.e	. . . . . . . . . . 11 ⊢ 𝐸 = (Edg‘𝐺)
14	13	eleq2i 2825	. . . . . . . . . 10 ⊢ ({(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))
15	14	anbi2i 623	. . . . . . . . 9 ⊢ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ↔ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
16	15	anbi1i 624	. . . . . . . 8 ⊢ ((((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
17	12, 16	bitr2i 276	. . . . . . 7 ⊢ ((((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋))
18	11, 17	anbi12i 628	. . . . . 6 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
19	7, 18	bitri 275	. . . . 5 ⊢ (((𝑤 ∈ Word (Vtx‘𝐺) ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
20	6, 19	bitri 275	. . . 4 ⊢ ((((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
21	4, 20	bitri 275	. . 3 ⊢ ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
22	21	rabbia2 3399	. 2 ⊢ {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}
23	2, 22	eqtri 2756	1 ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3396  {cpr 4579  ‘cfv 6489  (class class class)co 7355  0cc0 11017  1c1 11018  2c2 12191  ♯chash 14244  Word cword 14427  Vtxcvtx 28995  Edgcedg 29046   ClWWalksN cclwwlkn 30025  ClWWalksNOncclwwlknon 30088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-oadd 8398  df-er 8631  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9843  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-nn 12137  df-2 12199  df-n0 12393  df-xnn0 12466  df-z 12480  df-uz 12743  df-fz 13415  df-fzo 13562  df-hash 14245  df-word 14428  df-lsw 14477  df-clwwlk 29983  df-clwwlkn 30026  df-clwwlknon 30089

This theorem is referenced by:  s2elclwwlknon2  30105  clwwlknon2num  30106