Theorem clwwlknon2x 16220

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 16219	. 2 ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
3		clwwlkn2 16206	. . . . 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 1014	. . . . . 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 2233	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = 𝑉
10	9	wrdeqi 11123	. . . . . . . 8 ⊢ Word (Vtx‘𝐺) = Word 𝑉
11	10	eleq2i 2296	. . . . . . 7 ⊢ (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
12		df-3an 1004	. . . . . . . 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 2296	. . . . . . . . . 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 2785	. 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 2250	1 ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}

Syntax hints:   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {crab 2512  {cpr 3668  ‘cfv 5322  (class class class)co 6011  0cc0 8020  1c1 8021  2c2 9182  ♯chash 11025  Word cword 11100  Vtxcvtx 15850  Edgcedg 15895   ClWWalksN cclwwlkn 16188  ClWWalksNOncclwwlknon 16211

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-map 6812  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-inn 9132  df-2 9190  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-lsw 11146  df-ndx 13072  df-slot 13073  df-base 13075  df-vtx 15852  df-clwwlk 16177  df-clwwlkn 16189  df-clwwlknon 16212

This theorem is referenced by:  s2elclwwlknon2  16221