wlksnwwlknvbij - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wlksnwwlknvbij		Structured version Visualization version GIF version

Theorem wlksnwwlknvbij 29706

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 6904	. . . . 5 ⊢ (Walks‘𝐺) ∈ V
2	1	mptrabex 7231	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ∈ V
3	2	resex 6027	. . 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 2727	. . . 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 2727	. . . . 5 ⊢ {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁} = {𝑞 ∈ (Walks‘𝐺) ∣ (♯‘(1^st ‘𝑞)) = 𝑁}
6		eqid 2727	. . . . 5 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
7	5, 6, 4	wlknwwlksnbij 29686	. . . 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 6890	. . . . . 6 ⊢ (𝑤 = (2^nd ‘𝑝) → (𝑤‘0) = ((2^nd ‘𝑝)‘0))
9	8	eqeq1d 2729	. . . . 5 ⊢ (𝑤 = (2^nd ‘𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
10	9	3ad2ant3 1133	. . . 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 7130	. . 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 6821	. . . 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 3582	. . 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 6896	. . . . . . 7 ⊢ (𝑝 = 𝑞 → (♯‘(1^st ‘𝑝)) = (♯‘(1^st ‘𝑞)))
16	15	eqeq1d 2729	. . . . . 6 ⊢ (𝑝 = 𝑞 → ((♯‘(1^st ‘𝑝)) = 𝑁 ↔ (♯‘(1^st ‘𝑞)) = 𝑁))
17	16	rabrabi 3445	. . . . 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 2736	. . . 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 6822	. . . 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 1917	. 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 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {crab 3427 Vcvv 3469 ↦ cmpt 5225 ↾ cres 5674 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 0cc0 11130 ℕ₀cn0 12494 ♯chash 14313 USPGraphcuspgr 28948 Walkscwlks 29397 WWalksN cwwlksn 29624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-edg 28848 df-uhgr 28858 df-upgr 28882 df-uspgr 28950 df-wlks 29400 df-wwlks 29628 df-wwlksn 29629

This theorem is referenced by: rusgrnumwlkg 29775

W3C validator