Theorem clwwlknclwwlkdif 29954

Description: The set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑉 and ending not at this vertex is the difference between the set 𝐶 of walks of length 𝑁 starting with this vertex 𝑋 and the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 16-Mar-2022.)

Hypotheses

Ref	Expression
clwwlknclwwlkdif.a	⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
clwwlknclwwlkdif.b	⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
clwwlknclwwlkdif.c	⊢ 𝐶 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}

Assertion

Ref	Expression
clwwlknclwwlkdif	⊢ 𝐴 = (𝐶 ∖ 𝐵)

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑋

Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑤)   𝐶(𝑤)

Proof of Theorem clwwlknclwwlkdif

Step	Hyp	Ref	Expression
1		clwwlknclwwlkdif.a	. 2 ⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
2		clwwlknclwwlkdif.c	. . . 4 ⊢ 𝐶 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
3		clwwlknclwwlkdif.b	. . . . 5 ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)
4		eqid 2731	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	iswwlksnon 29829	. . . . 5 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
6	3, 5	eqtri 2754	. . . 4 ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
7	2, 6	difeq12i 4074	. . 3 ⊢ (𝐶 ∖ 𝐵) = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)})
8		difrab 4268	. . 3 ⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋))}
9		annotanannot 834	. . . . 5 ⊢ (((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)) ↔ ((𝑤‘0) = 𝑋 ∧ ¬ (𝑤‘𝑁) = 𝑋))
10		df-ne 2929	. . . . . . 7 ⊢ ((𝑤‘𝑁) ≠ 𝑋 ↔ ¬ (𝑤‘𝑁) = 𝑋)
11		wwlknlsw 29823	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤‘𝑁) = (lastS‘𝑤))
12	11	neeq1d 2987	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘𝑁) ≠ 𝑋 ↔ (lastS‘𝑤) ≠ 𝑋))
13	10, 12	bitr3id 285	. . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (¬ (𝑤‘𝑁) = 𝑋 ↔ (lastS‘𝑤) ≠ 𝑋))
14	13	anbi2d 630	. . . . 5 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ ¬ (𝑤‘𝑁) = 𝑋) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)))
15	9, 14	bitrid 283	. . . 4 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)))
16	15	rabbiia 3399	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋))} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}
17	7, 8, 16	3eqtrri 2759	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = (𝐶 ∖ 𝐵)
18	1, 17	eqtri 2754	1 ⊢ 𝐴 = (𝐶 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  {crab 3395   ∖ cdif 3899  ‘cfv 6481  (class class class)co 7346  0cc0 11003  lastSclsw 14466  Vtxcvtx 28972   WWalksN cwwlksn 29802   WWalksNOn cwwlksnon 29803

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405  df-fzo 13552  df-hash 14235  df-word 14418  df-lsw 14467  df-wwlks 29806  df-wwlksn 29807  df-wwlksnon 29808

This theorem is referenced by:  clwwlknclwwlkdifnum  29955