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Theorem wlknwwlksnsur 26645
Description: Lemma 3 for wlknwwlksnbij2 26647. (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 25964 . . 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 26643 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5sylan 488 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
73eleq2i 2690 . . . 4 (𝑝𝑊𝑝 ∈ (𝑁 WWalksN 𝐺))
8 wlklnwwlkn 26639 . . . . . . . . . . 11 (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
98adantr 481 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (∃𝑓(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ (𝑁 WWalksN 𝐺)))
10 df-br 4614 . . . . . . . . . . . 12 (𝑓(Walks‘𝐺)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺))
11 vex 3189 . . . . . . . . . . . . . . . 16 𝑓 ∈ V
12 vex 3189 . . . . . . . . . . . . . . . 16 𝑝 ∈ V
1311, 12op1st 7121 . . . . . . . . . . . . . . 15 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1413eqcomi 2630 . . . . . . . . . . . . . 14 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1514fveq2i 6151 . . . . . . . . . . . . 13 (#‘𝑓) = (#‘(1st ‘⟨𝑓, 𝑝⟩))
1615eqeq1i 2626 . . . . . . . . . . . 12 ((#‘𝑓) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
17 elex 3198 . . . . . . . . . . . . . 14 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ⟨𝑓, 𝑝⟩ ∈ V)
18 eleq1 2686 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (Walks‘𝐺) ↔ ⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺)))
1918biimparc 504 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑢 ∈ (Walks‘𝐺))
2019adantr 481 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑢 ∈ (Walks‘𝐺))
21 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
2221fveq2d 6152 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (#‘(1st𝑢)) = (#‘(1st ‘⟨𝑓, 𝑝⟩)))
2322eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → ((#‘(1st𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2423adantl 482 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((#‘(1st𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2524biimpar 502 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (#‘(1st𝑢)) = 𝑁)
26 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
2711, 12op2nd 7122 . . . . . . . . . . . . . . . . . . 19 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
2826, 27syl6req 2672 . . . . . . . . . . . . . . . . . 18 (𝑢 = ⟨𝑓, 𝑝⟩ → 𝑝 = (2nd𝑢))
2928adantl 482 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑝 = (2nd𝑢))
3029adantr 481 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑝 = (2nd𝑢))
3120, 25, 30jca31 556 . . . . . . . . . . . . . . 15 (((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3231ex 450 . . . . . . . . . . . . . 14 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3317, 32spcimedv 3278 . . . . . . . . . . . . 13 (⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3433imp 445 . . . . . . . . . . . 12 ((⟨𝑓, 𝑝⟩ ∈ (Walks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3510, 16, 34syl2anb 496 . . . . . . . . . . 11 ((𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3635exlimiv 1855 . . . . . . . . . 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 445 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
39 fveq2 6148 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
4039fveq2d 6152 . . . . . . . . . . . 12 (𝑝 = 𝑢 → (#‘(1st𝑝)) = (#‘(1st𝑢)))
4140eqeq1d 2623 . . . . . . . . . . 11 (𝑝 = 𝑢 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑢)) = 𝑁))
4241elrab 3346 . . . . . . . . . 10 (𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁} ↔ (𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁))
4342anbi1i 730 . . . . . . . . 9 ((𝑢 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4443exbii 1771 . . . . . . . 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 2913 . . . . . . 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 3132 . . . . . 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 6148 . . . . . . . 8 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
51 fvex 6158 . . . . . . . 8 (2nd𝑢) ∈ V
5250, 4, 51fvmpt 6239 . . . . . . 7 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
5352eqeq2d 2631 . . . . . 6 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
5453rexbiia 3033 . . . . 5 (∃𝑢𝑇 𝑝 = (𝐹𝑢) ↔ ∃𝑢𝑇 𝑝 = (2nd𝑢))
5549, 54sylibr 224 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
567, 55sylan2b 492 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
5756ralrimiva 2960 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
58 dffo3 6330 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
596, 57, 58sylanbrc 697 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cop 4154   class class class wbr 4613  cmpt 4673  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cn0 11236  #chash 13057   UPGraph cupgr 25871   USPGraph cuspgr 25936  Walkscwlks 26362   WWalksN cwwlksn 26587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-edg 25840  df-uhgr 25849  df-upgr 25873  df-uspgr 25938  df-wlks 26365  df-wwlks 26591  df-wwlksn 26592
This theorem is referenced by:  wlknwwlksnbij  26646
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