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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  ontowfo 6099  cfv 6101  (class class class)co 6878  1st c1st 7399  2nd c2nd 7400  0cn0 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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