Theorem clwwlknclwwlkdif 30271

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 2769	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	iswwlksnon 30143	. . . . 5 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
6	3, 5	eqtri 2792	. . . 4 ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
7	2, 6	difeq12i 4087	. . 3 ⊢ (𝐶 ∖ 𝐵) = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)})
8		difrab 4279	. . 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 847	. . . . 5 ⊢ (((𝑤‘0) = 𝑋 ∧ ¬ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)) ↔ ((𝑤‘0) = 𝑋 ∧ ¬ (𝑤‘𝑁) = 𝑋))
10		df-ne 2965	. . . . . . 7 ⊢ ((𝑤‘𝑁) ≠ 𝑋 ↔ ¬ (𝑤‘𝑁) = 𝑋)
11		wwlknlsw 30137	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤‘𝑁) = (lastS‘𝑤))
12	11	neeq1d 3023	. . . . . . 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 641	. . . . 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 3427	. . 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 2797	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = (𝐶 ∖ 𝐵)
18	1, 17	eqtri 2792	1 ⊢ 𝐴 = (𝐶 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  {crab 3423   ∖ cdif 3910  ‘cfv 6537  (class class class)co 7411  0cc0 11100  lastSclsw 14599  Vtxcvtx 29287   WWalksN cwwlksn 30116   WWalksNOn cwwlksnon 30117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-hash 14367  df-word 14551  df-lsw 14600  df-wwlks 30120  df-wwlksn 30121  df-wwlksnon 30122

This theorem is referenced by:  clwwlknclwwlkdifnum  30272