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Theorem wlknwwlksnsur 26999
Description: Lemma 3 for wlknwwlksnbij2 27001. (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 26270 . . 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 26997 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5sylan 489 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
73eleq2i 2831 . . . 4 (𝑝𝑊𝑝 ∈ (𝑁 WWalksN 𝐺))
8 wlklnwwlkn 26993 . . . . . . . . . . 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 4805 . . . . . . . . . . . 12 (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
11 vex 3343 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
12 vex 3343 . . . . . . . . . . . . . . . 16 𝑝 ∈ V
1311, 12op1st 7341 . . . . . . . . . . . . . . 15 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1413eqcomi 2769 . . . . . . . . . . . . . 14 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1514fveq2i 6355 . . . . . . . . . . . . 13 (♯‘𝑓) = (♯‘(1st ‘⟨𝑓, 𝑝⟩))
1615eqeq1i 2765 . . . . . . . . . . . 12 ((♯‘𝑓) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
17 elex 3352 . . . . . . . . . . . . . 14 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ⟨𝑓, 𝑝⟩ ∈ V)
18 eleq1 2827 . . . . . . . . . . . . . . . . . 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 6352 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
2221fveq2d 6356 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (♯‘(1st𝑢)) = (♯‘(1st ‘⟨𝑓, 𝑝⟩)))
2322eqeq1d 2762 . . . . . . . . . . . . . . . . . 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 6352 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
2711, 12op2nd 7342 . . . . . . . . . . . . . . . . . . 19 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
2826, 27syl6req 2811 . . . . . . . . . . . . . . . . . 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 3432 . . . . . . . . . . . . 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 2007 . . . . . . . . . 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 6352 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4039fveq2d 6356 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (♯‘(1st𝑝)) = (♯‘(1st𝑢)))
4140eqeq1d 2762 . . . . . . . . . . 11 (𝑝 = 𝑢 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑢)) = 𝑁))
4241elrab 3504 . . . . . . . . . 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 1923 . . . . . . . 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 3056 . . . . . . 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 3282 . . . . . 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 6352 . . . . . . . 8 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
51 fvex 6362 . . . . . . . 8 (2nd𝑢) ∈ V
5250, 4, 51fvmpt 6444 . . . . . . 7 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
5352eqeq2d 2770 . . . . . 6 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
5453rexbiia 3178 . . . . 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 3104 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
58 dffo3 6537 . 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 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cop 4327   class class class wbr 4804  cmpt 4881  wf 6045  ontowfo 6047  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  0cn0 11484  chash 13311  UPGraphcupgr 26174  USPGraphcuspgr 26242  Walkscwlks 26702   WWalksN cwwlksn 26929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
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 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-edg 26139  df-uhgr 26152  df-upgr 26176  df-uspgr 26244  df-wlks 26705  df-wwlks 26933  df-wwlksn 26934
This theorem is referenced by:  wlknwwlksnbij  27000
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