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Theorem wlknwwlksnbij 27242
 Description: The mapping (𝑡 ∈ 𝑇 ↦ (2nd ‘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) ∣ (♯‘(1st𝑝)) = 𝑁}
wlknwwlksnbij.w 𝑊 = (𝑁 WWalksN 𝐺)
wlknwwlksnbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlknwwlksnbij ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)
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.
StepHypRef Expression
1 eqid 2778 . . 3 (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
21wlkswwlksf1o 27228 . . . 4 (𝐺 ∈ USPGraph → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
32adantr 474 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)):(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
4 fveqeq2 6455 . . . . 5 (𝑞 = (2nd𝑝) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
543ad2ant3 1126 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(2nd𝑝)) = (𝑁 + 1)))
6 wlkcpr 26976 . . . . . . 7 (𝑝 ∈ (Walks‘𝐺) ↔ (1st𝑝)(Walks‘𝐺)(2nd𝑝))
7 wlklenvp1 26966 . . . . . . . 8 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1))
8 eqeq1 2782 . . . . . . . . . 10 ((♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ ((♯‘(1st𝑝)) + 1) = (𝑁 + 1)))
9 wlkcl 26963 . . . . . . . . . . . . 13 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℕ0)
109nn0cnd 11704 . . . . . . . . . . . 12 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → (♯‘(1st𝑝)) ∈ ℂ)
1110adantr 474 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (♯‘(1st𝑝)) ∈ ℂ)
12 nn0cn 11653 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
1312adantl 475 . . . . . . . . . . . 12 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ)
1413adantl 475 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 𝑁 ∈ ℂ)
15 1cnd 10371 . . . . . . . . . . 11 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → 1 ∈ ℂ)
1611, 14, 15addcan2d 10580 . . . . . . . . . 10 (((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) → (((♯‘(1st𝑝)) + 1) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
178, 16sylan9bbr 506 . . . . . . . . 9 ((((1st𝑝)(Walks‘𝐺)(2nd𝑝) ∧ (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0)) ∧ (♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1)) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
1817exp31 412 . . . . . . . 8 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd𝑝)) = ((♯‘(1st𝑝)) + 1) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))))
197, 18mpid 44 . . . . . . 7 ((1st𝑝)(Walks‘𝐺)(2nd𝑝) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁)))
206, 19sylbi 209 . . . . . 6 (𝑝 ∈ (Walks‘𝐺) → ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁)))
2120impcom 398 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺)) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
22213adant3 1123 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘(2nd𝑝)) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
235, 22bitrd 271 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (Walks‘𝐺) ∧ 𝑞 = (2nd𝑝)) → ((♯‘𝑞) = (𝑁 + 1) ↔ (♯‘(1st𝑝)) = 𝑁))
241, 3, 23f1oresrab 6659 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
25 wlknwwlksnbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
26 wlknwwlksnbij.t . . . . . . 7 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}
2726mpteq1i 4974 . . . . . 6 (𝑡𝑇 ↦ (2nd𝑡)) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡))
28 ssrab2 3908 . . . . . . 7 {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ⊆ (Walks‘𝐺)
29 resmpt 5699 . . . . . . 7 ({𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ⊆ (Walks‘𝐺) → ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡)))
3028, 29ax-mp 5 . . . . . 6 ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = (𝑡 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↦ (2nd𝑡))
31 fveq2 6446 . . . . . . . 8 (𝑡 = 𝑝 → (2nd𝑡) = (2nd𝑝))
3231cbvmptv 4985 . . . . . . 7 (𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) = (𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝))
3332reseq1i 5638 . . . . . 6 ((𝑡 ∈ (Walks‘𝐺) ↦ (2nd𝑡)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
3427, 30, 333eqtr2i 2808 . . . . 5 (𝑡𝑇 ↦ (2nd𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
3534a1i 11 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑡𝑇 ↦ (2nd𝑡)) = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}))
3625, 35syl5eq 2826 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹 = ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}))
3726a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁})
38 wlknwwlksnbij.w . . . 4 𝑊 = (𝑁 WWalksN 𝐺)
39 wwlksn 27186 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4039adantl 475 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4138, 40syl5eq 2826 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝑊 = {𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)})
4236, 37, 41f1oeq123d 6386 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝐹:𝑇1-1-onto𝑊 ↔ ((𝑝 ∈ (Walks‘𝐺) ↦ (2nd𝑝)) ↾ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}):{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}–1-1-onto→{𝑞 ∈ (WWalks‘𝐺) ∣ (♯‘𝑞) = (𝑁 + 1)}))
4324, 42mpbird 249 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)
 Colors of variables: wff setvar class 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  1st c1st 7443  2nd 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
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