Theorem clwwlknclwwlkdif 30457

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 2762	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	iswwlksnon 30329	. . . . 5 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
6	3, 5	eqtri 2785	. . . 4 ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
7	2, 6	difeq12i 4075	. . 3 ⊢ (𝐶 ∖ 𝐵) = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)})
8		difrab 4267	. . 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 848	. . . . 5 ⊢ (((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)) ↔ ((𝑤‘0) = 𝑋 ∧ ¬ (𝑤‘𝑁) = 𝑋))
10		df-ne 2958	. . . . . . 7 ⊢ ((𝑤‘𝑁) ≠ 𝑋 ↔ ¬ (𝑤‘𝑁) = 𝑋)
11		wwlknlsw 30323	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤‘𝑁) = (lastS‘𝑤))
12	11	neeq1d 3016	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘𝑁) ≠ 𝑋 ↔ (lastS‘𝑤) ≠ 𝑋))
13	10, 12	bitr3id 288	. . . . . 6 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (¬ (𝑤‘𝑁) = 𝑋 ↔ (lastS‘𝑤) ≠ 𝑋))
14	13	anbi2d 642	. . . . 5 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ ¬ (𝑤‘𝑁) = 𝑋) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)))
15	9, 14	bitrid 286	. . . 4 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)))
16	15	rabbiia 3418	. . 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 2790	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = (𝐶 ∖ 𝐵)
18	1, 17	eqtri 2785	1 ⊢ 𝐴 = (𝐶 ∖ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {crab 3414   ∖ cdif 3899  ‘cfv 6537  (class class class)co 7417  0cc0 11128  lastSclsw 14631  Vtxcvtx 29461   WWalksN cwwlksn 30302   WWalksNOn cwwlksnon 30303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-hash 14399  df-word 14583  df-lsw 14632  df-wwlks 30306  df-wwlksn 30307  df-wwlksnon 30308

This theorem is used by:  clwwlknclwwlkdifnum  30458