Theorem wlknwwlksnbij 27242

Description: The mapping (𝑡 ∈ 𝑇 ↦ (2^nd ‘𝑡)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)

Hypotheses

Ref	Expression
wlknwwlksnbij.t	⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}
wlknwwlksnbij.w	⊢ 𝑊 = (𝑁 WWalksN 𝐺)
wlknwwlksnbij.f	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))

Assertion

Ref	Expression
wlknwwlksnbij	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

Distinct variable groups:   𝐺,𝑝,𝑡   𝑁,𝑝,𝑡   𝑡,𝑇

Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑡,𝑝)   𝑊(𝑡,𝑝)

Proof of Theorem wlknwwlksnbij

Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2778	. . 3 ⊢ (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
2	1	wlkswwlksf1o 27228	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
3	2	adantr 474	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4		fveqeq2 6455	. . . . 5 ⊢ (𝑞 = (2nd ‘𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
5	4	3ad2ant3 1126	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd ‘𝑝)) = (𝑁 + 1)))
6		wlkcpr 26976	. . . . . . 7 ⊢ (𝑝 ∈ (Walks‘𝐺) ↔ (1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝))
7		wlklenvp1 26966	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1))
8		eqeq1 2782	. . . . . . . . . 10 ⊢ ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1)))
9		wlkcl 26963	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℕ0)
10	9	nn0cnd 11704	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → (♯‘(1st ‘𝑝)) ∈ ℂ)
11	10	adantr 474	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st ‘𝑝)) ∈ ℂ)
12		nn0cn 11653	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
13	12	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ)
14	13	adantl 475	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
15		1cnd 10371	. . . . . . . . . . 11 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
16	11, 14, 15	addcan2d 10580	. . . . . . . . . 10 ⊢ (((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (((♯‘(1st ‘𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
17	8, 16	sylan9bbr 506	. . . . . . . . 9 ⊢ ((((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) ∧ (♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
18	17	exp31 412	. . . . . . . 8 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = ((♯‘(1st ‘𝑝)) + 1) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))))
19	7, 18	mpid 44	. . . . . . 7 ⊢ ((1st ‘𝑝)(Walks‘𝐺)(2nd ‘𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
20	6, 19	sylbi 209	. . . . . 6 ⊢ (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁)))
21	20	impcom 398	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
22	21	3adant3 1123	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘(2nd ‘𝑝)) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
23	5, 22	bitrd 271	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd ‘𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st ‘𝑝)) = 𝑁))
24	1, 3, 23	f1oresrab 6659	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
25		wlknwwlksnbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
26		wlknwwlksnbij.t	. . . . . . 7 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}
27	26	mpteq1i 4974	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
28		ssrab2 3908	. . . . . . 7 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29		resmpt 5699	. . . . . . 7 ⊢ ({𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ⊆ (Walks‘𝐺) → ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡)))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ↦ (2nd ‘𝑡))
31		fveq2 6446	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝))
32	31	cbvmptv 4985	. . . . . . 7 ⊢ (𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝))
33	32	reseq1i 5638	. . . . . 6 ⊢ ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁})
34	27, 30, 33	3eqtr2i 2808	. . . . 5 ⊢ (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁})
35	34	a1i 11	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}))
36	25, 35	syl5eq 2826	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹 = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}))
37	26	a1i 11	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁})
38		wlknwwlksnbij.w	. . . 4 ⊢ 𝑊 = (𝑁 WWalksN 𝐺)
39		wwlksn 27186	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
40	39	adantl 475	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
41	38, 40	syl5eq 2826	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
42	36, 37, 41	f1oeq123d 6386	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝐹:𝑇–1-1-onto→𝑊 ↔ ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)}))
43	24, 42	mpbird 249	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {crab 3094   ⊆ wss 3792   class class class wbr 4886   ↦ cmpt 4965   ↾ cres 5357  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922  1^st c1st 7443  2^nd c2nd 7444  ℂcc 10270  1c1 10273   + caddc 10275  ℕ₀cn0 11642  ♯chash 13435  USPGraphcuspgr 26497  Walkscwlks 26944  WWalkscwwlks 27174   WWalksN cwwlksn 27175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ifp 1047  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-edg 26396  df-uhgr 26406  df-upgr 26430  df-uspgr 26499  df-wlks 26947  df-wwlks 27179  df-wwlksn 27180

This theorem is referenced by:  wlknwwlksnbij2OLD  27243  wlknwwlksnen  27244  wlksnwwlknvbij  27281