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Theorem wlknwwlksnsur 26995
Description: Lemma 3 for wlknwwlksnbij2 26997. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
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
wlknwwlksnsur ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Distinct variable groups:   𝐺,𝑝,𝑡   𝑁,𝑝,𝑡   𝑡,𝑇   𝑡,𝑊   𝐹,𝑝   𝑇,𝑝   𝑊,𝑝
Allowed substitution hint:   𝐹(𝑡)

Proof of Theorem wlknwwlksnsur
Dummy variables 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgrupgr 26266 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2 wlknwwlksnbij.t . . . 4 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}
3 wlknwwlksnbij.w . . . 4 𝑊 = (𝑁 WWalksN 𝐺)
4 wlknwwlksnbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
52, 3, 4wlknwwlksnfun 26993 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5sylan 489 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
73eleq2i 2827 . . . 4 (𝑝𝑊𝑝 ∈ (𝑁 WWalksN 𝐺))
8 wlklnwwlkn 26989 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
98adantr 472 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
10 df-br 4801 . . . . . . . . . . . 12 (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
11 vex 3339 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
12 vex 3339 . . . . . . . . . . . . . . . 16 𝑝 ∈ V
1311, 12op1st 7337 . . . . . . . . . . . . . . 15 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1413eqcomi 2765 . . . . . . . . . . . . . 14 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1514fveq2i 6351 . . . . . . . . . . . . 13 (♯‘𝑓) = (♯‘(1st ‘⟨𝑓, 𝑝⟩))
1615eqeq1i 2761 . . . . . . . . . . . 12 ((♯‘𝑓) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
17 elex 3348 . . . . . . . . . . . . . 14 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ⟨𝑓, 𝑝⟩ ∈ V)
18 eleq1 2823 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
1918biimparc 505 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑢 ∈ (Walks‘𝐺))
2019adantr 472 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑢 ∈ (Walks‘𝐺))
21 fveq2 6348 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
2221fveq2d 6352 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (♯‘(1st𝑢)) = (♯‘(1st ‘⟨𝑓, 𝑝⟩)))
2322eqeq1d 2758 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → ((♯‘(1st𝑢)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2423adantl 473 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((♯‘(1st𝑢)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2524biimpar 503 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (♯‘(1st𝑢)) = 𝑁)
26 fveq2 6348 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
2711, 12op2nd 7338 . . . . . . . . . . . . . . . . . . 19 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
2826, 27syl6req 2807 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → 𝑝 = (2nd𝑢))
2928adantl 473 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑝 = (2nd𝑢))
3029adantr 472 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑝 = (2nd𝑢))
3120, 25, 30jca31 558 . . . . . . . . . . . . . . 15 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3231ex 449 . . . . . . . . . . . . . 14 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3317, 32spcimedv 3428 . . . . . . . . . . . . 13 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3433imp 444 . . . . . . . . . . . 12 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3510, 16, 34syl2anb 497 . . . . . . . . . . 11 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3635exlimiv 2003 . . . . . . . . . 10 (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
379, 36syl6bir 244 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (𝑁 WWalksN 𝐺) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3837imp 444 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
39 fveq2 6348 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4039fveq2d 6352 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (♯‘(1st𝑝)) = (♯‘(1st𝑢)))
4140eqeq1d 2758 . . . . . . . . . . 11 (𝑝 = 𝑢 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑢)) = 𝑁))
4241elrab 3500 . . . . . . . . . 10 (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁))
4342anbi1i 733 . . . . . . . . 9 ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4443exbii 1919 . . . . . . . 8 (∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4538, 44sylibr 224 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)))
46 df-rex 3052 . . . . . . 7 (∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)))
4745, 46sylibr 224 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢))
482rexeqi 3278 . . . . . 6 (∃𝑢𝑇 𝑝 = (2nd𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢))
4947, 48sylibr 224 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢𝑇 𝑝 = (2nd𝑢))
50 fveq2 6348 . . . . . . . 8 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
51 fvex 6358 . . . . . . . 8 (2nd𝑢) ∈ V
5250, 4, 51fvmpt 6440 . . . . . . 7 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
5352eqeq2d 2766 . . . . . 6 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
5453rexbiia 3174 . . . . 5 (∃𝑢𝑇 𝑝 = (𝐹𝑢) ↔ ∃𝑢𝑇 𝑝 = (2nd𝑢))
5549, 54sylibr 224 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
567, 55sylan2b 493 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
5756ralrimiva 3100 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
58 dffo3 6533 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
596, 57, 58sylanbrc 701 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1628  wex 1849  wcel 2135  wral 3046  wrex 3047  {crab 3050  Vcvv 3336  cop 4323   class class class wbr 4800  cmpt 4877  wf 6041  ontowfo 6043  cfv 6045  (class class class)co 6809  1st c1st 7327  2nd c2nd 7328  0cn0 11480  chash 13307  UPGraphcupgr 26170  USPGraphcuspgr 26238  Walkscwlks 26698   WWalksN cwwlksn 26925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-2o 7726  df-oadd 7729  df-er 7907  df-map 8021  df-pm 8022  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-card 8951  df-cda 9178  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-2 11267  df-n0 11481  df-xnn0 11552  df-z 11566  df-uz 11876  df-fz 12516  df-fzo 12656  df-hash 13308  df-word 13481  df-edg 26135  df-uhgr 26148  df-upgr 26172  df-uspgr 26240  df-wlks 26701  df-wwlks 26929  df-wwlksn 26930
This theorem is referenced by:  wlknwwlksnbij  26996
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