Theorem clwwlknclwwlkdif 29941

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 2729	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	iswwlksnon 29816	. . . . 5 ⊢ (𝑋(𝑁 WWalksNOn 𝐺)𝑋) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
6	3, 5	eqtri 2752	. . . 4 ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)}
7	2, 6	difeq12i 4077	. . 3 ⊢ (𝐶 ∖ 𝐵) = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘𝑁) = 𝑋)})
8		difrab 4271	. . 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 2926	. . . . . . 7 ⊢ ((𝑤‘𝑁) ≠ 𝑋 ↔ ¬ (𝑤‘𝑁) = 𝑋)
11		wwlknlsw 29810	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (𝑤‘𝑁) = (lastS‘𝑤))
12	11	neeq1d 2984	. . . . . . 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 3400	. . 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 2757	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} = (𝐶 ∖ 𝐵)
18	1, 17	eqtri 2752	1 ⊢ 𝐴 = (𝐶 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3396   ∖ cdif 3902  ‘cfv 6486  (class class class)co 7353  0cc0 11028  lastSclsw 14487  Vtxcvtx 28959   WWalksN cwwlksn 29789   WWalksNOn cwwlksnon 29790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-hash 14256  df-word 14439  df-lsw 14488  df-wwlks 29793  df-wwlksn 29794  df-wwlksnon 29795

This theorem is referenced by:  clwwlknclwwlkdifnum  29942