Theorem clwwlknon2x 16359

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 16358	. 2 ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
3		clwwlkn2 16345	. . . . 5 ⊢ (𝑤 ∈ (2 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 2 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
4	3	anbi1i 458	. . . 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 1017	. . . . . 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 458	. . . . 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 401	. . . . . 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 2235	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 11185	. . . . . . . 8 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2298	. . . . . . 7 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12		df-3an 1007	. . . . . . . 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 2298	. . . . . . . . . 10 ⊢ ({(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))
15	14	anbi2i 457	. . . . . . . . 9 ⊢ (((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸) ↔ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺)))
16	15	anbi1i 458	. . . . . . . 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 185	. . . . . . 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 460	. . . . . 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 184	. . . . 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 184	. . . 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 184	. . 3 ⊢ ((𝑤 ∈ (2 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)))
22	21	rabbia2 2788	. 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 2252	1 ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}

Syntax hints:   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  {crab 2515  {cpr 3674  ‘cfv 5333  (class class class)co 6028  0cc0 8075  1c1 8076  2c2 9236  ♯chash 11083  Word cword 11162  Vtxcvtx 15936  Edgcedg 15981   ClWWalksN cclwwlkn 16327  ClWWalksNOncclwwlknon 16350

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-inn 9186  df-2 9244  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-lsw 11208  df-ndx 13148  df-slot 13149  df-base 13151  df-vtx 15938  df-clwwlk 16316  df-clwwlkn 16328  df-clwwlknon 16351

This theorem is referenced by:  s2elclwwlknon2  16360