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Theorem wlknwwlksnsurOLD 27148
 Description: Obsolete as of 5-Aug-2022. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
wlknwwlksnbijOLD.t 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}
wlknwwlksnbijOLD.w 𝑊 = (𝑁 WWalksN 𝐺)
wlknwwlksnbijOLD.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlknwwlksnsurOLD ((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 wlknwwlksnsurOLD
Dummy variables 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgrupgr 26412 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2 wlknwwlksnbijOLD.t . . . 4 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}
3 wlknwwlksnbijOLD.w . . . 4 𝑊 = (𝑁 WWalksN 𝐺)
4 wlknwwlksnbijOLD.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
52, 3, 4wlknwwlksnfunOLD 27146 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5sylan 576 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
73eleq2i 2870 . . . 4 (𝑝𝑊𝑝 ∈ (𝑁 WWalksN 𝐺))
8 wlklnwwlkn 27142 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
98adantr 473 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
10 df-br 4844 . . . . . . . . . . . 12 (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
11 vex 3388 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
12 vex 3388 . . . . . . . . . . . . . . . 16 𝑝 ∈ V
1311, 12op1st 7409 . . . . . . . . . . . . . . 15 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1413eqcomi 2808 . . . . . . . . . . . . . 14 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1514fveq2i 6414 . . . . . . . . . . . . 13 (♯‘𝑓) = (♯‘(1st ‘⟨𝑓, 𝑝⟩))
1615eqeq1i 2804 . . . . . . . . . . . 12 ((♯‘𝑓) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
17 elex 3400 . . . . . . . . . . . . . 14 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ⟨𝑓, 𝑝⟩ ∈ V)
18 eleq1 2866 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
1918biimparc 472 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑢 ∈ (Walks‘𝐺))
2019adantr 473 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑢 ∈ (Walks‘𝐺))
21 fveq2 6411 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
2221fveq2d 6415 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (♯‘(1st𝑢)) = (♯‘(1st ‘⟨𝑓, 𝑝⟩)))
2322eqeq1d 2801 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → ((♯‘(1st𝑢)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2423adantl 474 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((♯‘(1st𝑢)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2524biimpar 470 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (♯‘(1st𝑢)) = 𝑁)
26 fveq2 6411 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
2711, 12op2nd 7410 . . . . . . . . . . . . . . . . . . 19 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
2826, 27syl6req 2850 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → 𝑝 = (2nd𝑢))
2928adantl 474 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑝 = (2nd𝑢))
3029adantr 473 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑝 = (2nd𝑢))
3120, 25, 30jca31 511 . . . . . . . . . . . . . . 15 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3231ex 402 . . . . . . . . . . . . . 14 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3317, 32spcimedv 3480 . . . . . . . . . . . . 13 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ((♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3433imp 396 . . . . . . . . . . . 12 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3510, 16, 34syl2anb 592 . . . . . . . . . . 11 ((𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3635exlimiv 2026 . . . . . . . . . 10 (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
379, 36syl6bir 246 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (𝑝 ∈ (𝑁 WWalksN 𝐺) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3837imp 396 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
39 fveq2 6411 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4039fveq2d 6415 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (♯‘(1st𝑝)) = (♯‘(1st𝑢)))
4140eqeq1d 2801 . . . . . . . . . . 11 (𝑝 = 𝑢 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑢)) = 𝑁))
4241elrab 3556 . . . . . . . . . 10 (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁))
4342anbi1i 618 . . . . . . . . 9 ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4443exbii 1944 . . . . . . . 8 (∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4538, 44sylibr 226 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)))
46 df-rex 3095 . . . . . . 7 (∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)))
4745, 46sylibr 226 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢))
482rexeqi 3326 . . . . . 6 (∃𝑢𝑇 𝑝 = (2nd𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢))
4947, 48sylibr 226 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢𝑇 𝑝 = (2nd𝑢))
50 fveq2 6411 . . . . . . . 8 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
51 fvex 6424 . . . . . . . 8 (2nd𝑢) ∈ V
5250, 4, 51fvmpt 6507 . . . . . . 7 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
5352eqeq2d 2809 . . . . . 6 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
5453rexbiia 3221 . . . . 5 (∃𝑢𝑇 𝑝 = (𝐹𝑢) ↔ ∃𝑢𝑇 𝑝 = (2nd𝑢))
5549, 54sylibr 226 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
567, 55sylan2b 588 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
5756ralrimiva 3147 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
58 dffo3 6600 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
596, 57, 58sylanbrc 579 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653  ∃wex 1875   ∈ wcel 2157  ∀wral 3089  ∃wrex 3090  {crab 3093  Vcvv 3385  ⟨cop 4374   class class class wbr 4843   ↦ cmpt 4922  ⟶wf 6097  –onto→wfo 6099  ‘cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  ℕ0cn0 11580  ♯chash 13370  UPGraphcupgr 26315  USPGraphcuspgr 26384  Walkscwlks 26846   WWalksN cwwlksn 27077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-cda 9278  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-edg 26283  df-uhgr 26293  df-upgr 26317  df-uspgr 26386  df-wlks 26849  df-wwlks 27081  df-wwlksn 27082 This theorem is referenced by:  wlknwwlksnbijOLD  27149
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