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Theorem wlksnwwlknvbij 30197

Description: There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 5-Aug-2022.)

**Assertion**
Ref	Expression
wlksnwwlknvbij	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

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

Proof of Theorem wlksnwwlknvbij Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6895	. . . . 5 ⊢ (Walks‘𝐺) ∈ V
2	1	mptrabex 7224	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ∈ V
3	2	resex 6029	. . 3 ⊢ ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) ∈ V
4		eqid 2769	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) = (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝))
5		eqid 2769	. . . . 5 ⊢ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} = {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁}
6		eqid 2769	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
7	5, 6, 4	wlknwwlksnbij 30177	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
8		fveq1 6881	. . . . . 6 ⊢ (𝑤 = (2^nd ‘𝑝) → (𝑤‘0) = ((2^nd ‘𝑝)‘0))
9	8	eqeq1d 2771	. . . . 5 ⊢ (𝑤 = (2^nd ‘𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
10	9	3ad2ant3 1151	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) ∧ 𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∧ 𝑤 = (2^nd ‘𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
11	4, 7, 10	f1oresrab 7124	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
12		f1oeq1 6809	. . . 4 ⊢ (𝑓 = ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) → (𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
13	12	spcegv 3565	. . 3 ⊢ (((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) ∈ V → (((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
14	3, 11, 13	mpsyl 69	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
15		2fveq3 6887	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (♯‘(1^st ‘𝑝)) = (♯‘(1^st ‘𝑞)))
16	15	eqeq1d 2771	. . . . . 6 ⊢ (𝑝 = 𝑞 → ((♯‘(1^st ‘𝑝)) = 𝑁 ↔ (♯‘(1^st ‘𝑞)) = 𝑁))
17	16	rabrabi 3442	. . . . 5 ⊢ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋} = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
18	17	eqcomi 2778	. . . 4 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}
19		f1oeq2 6810	. . . 4 ⊢ ({𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋} → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
20	18, 19	mp1i 14	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
21	20	exbidv 1948	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
22	14, 21	mpbird 260	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∃wex 1806 ∈ wcel 2149 {crab 3423 Vcvv 3463 ↦ cmpt 5196 ↾ cres 5664 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 1^st c1st 7983 2^nd c2nd 7984 0cc0 11099 ℕ₀cn0 12503 ♯chash 14365 USPGraphcuspgr 29438 Walkscwlks 29886 WWalksN cwwlksn 30115

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 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1077 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 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-n0 12504 df-xnn0 12577 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13682 df-hash 14366 df-word 14550 df-edg 29338 df-uhgr 29348 df-upgr 29372 df-uspgr 29440 df-wlks 29889 df-wwlks 30119 df-wwlksn 30120

This theorem is referenced by: rusgrnumwlkg 30269

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