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Theorem wlksnwwlknvbij 29895

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 6894	. . . . 5 ⊢ (Walks‘𝐺) ∈ V
2	1	mptrabex 7222	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ∈ V
3	2	resex 6021	. . 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 2736	. . . 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 2736	. . . . 5 ⊢ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} = {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁}
6		eqid 2736	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
7	5, 6, 4	wlknwwlksnbij 29875	. . . 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 6880	. . . . . 6 ⊢ (𝑤 = (2^nd ‘𝑝) → (𝑤‘0) = ((2^nd ‘𝑝)‘0))
9	8	eqeq1d 2738	. . . . 5 ⊢ (𝑤 = (2^nd ‘𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
10	9	3ad2ant3 1135	. . . 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 7122	. . 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 6811	. . . 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 3581	. . 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 68	. 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 6886	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (♯‘(1^st ‘𝑝)) = (♯‘(1^st ‘𝑞)))
16	15	eqeq1d 2738	. . . . . 6 ⊢ (𝑝 = 𝑞 → ((♯‘(1^st ‘𝑝)) = 𝑁 ↔ (♯‘(1^st ‘𝑞)) = 𝑁))
17	16	rabrabi 3440	. . . . 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 2745	. . . 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 6812	. . . 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 13	. . 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 1921	. 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 257	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 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 {crab 3420 Vcvv 3464 ↦ cmpt 5206 ↾ cres 5661 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 0cc0 11134 ℕ₀cn0 12506 ♯chash 14353 USPGraphcuspgr 29132 Walkscwlks 29581 WWalksN cwwlksn 29813

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 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 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8724 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9920 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-xnn0 12580 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-hash 14354 df-word 14537 df-edg 29032 df-uhgr 29042 df-upgr 29066 df-uspgr 29134 df-wlks 29584 df-wwlks 29817 df-wwlksn 29818

This theorem is referenced by: rusgrnumwlkg 29964

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