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Theorem wlknwwlknsur 26240
 Description: Lemma 3 for wlknwwlknbij2 26242. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
Hypotheses
Ref Expression
wlknwwlknbij.t 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}
wlknwwlknbij.w 𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)
wlknwwlknbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlknwwlknsur ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
Distinct variable groups:   𝐸,𝑝   𝑁,𝑝,𝑡   𝑡,𝑇   𝑉,𝑝   𝑡,𝑊   𝐹,𝑝   𝑇,𝑝   𝑊,𝑝
Allowed substitution hints:   𝐸(𝑡)   𝐹(𝑡)   𝑉(𝑡)

Proof of Theorem wlknwwlknsur
Dummy variables 𝑓 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlknwwlknbij.t . . . 4 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}
2 wlknwwlknbij.w . . . 4 𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)
3 wlknwwlknbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
41, 2, 3wlknwwlknfun 26238 . . 3 (𝑁 ∈ ℕ0𝐹:𝑇𝑊)
54adantl 481 . 2 ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
62eleq2i 2680 . . . . 5 (𝑝𝑊𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
7 wlklniswwlkn 26229 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) ↔ 𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
8 df-br 4584 . . . . . . . . . . . . 13 (𝑓(𝑉 Walks 𝐸)𝑝 ↔ ⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸))
9 vex 3176 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
10 vex 3176 . . . . . . . . . . . . . . . . 17 𝑝 ∈ V
119, 10op1st 7067 . . . . . . . . . . . . . . . 16 (1st ‘⟨𝑓, 𝑝⟩) = 𝑓
1211eqcomi 2619 . . . . . . . . . . . . . . 15 𝑓 = (1st ‘⟨𝑓, 𝑝⟩)
1312fveq2i 6106 . . . . . . . . . . . . . 14 (#‘𝑓) = (#‘(1st ‘⟨𝑓, 𝑝⟩))
1413eqeq1i 2615 . . . . . . . . . . . . 13 ((#‘𝑓) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁)
15 elex 3185 . . . . . . . . . . . . . . 15 (⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) → ⟨𝑓, 𝑝⟩ ∈ V)
16 eleq1 2676 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → (𝑢 ∈ (𝑉 Walks 𝐸) ↔ ⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸)))
1716biimparc 503 . . . . . . . . . . . . . . . . . 18 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑢 ∈ (𝑉 Walks 𝐸))
1817adantr 480 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑢 ∈ (𝑉 Walks 𝐸))
19 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = ⟨𝑓, 𝑝⟩ → (1st𝑢) = (1st ‘⟨𝑓, 𝑝⟩))
2019fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (#‘(1st𝑢)) = (#‘(1st ‘⟨𝑓, 𝑝⟩)))
2120eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → ((#‘(1st𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2221adantl 481 . . . . . . . . . . . . . . . . . 18 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((#‘(1st𝑢)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁))
2322biimpar 501 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → (#‘(1st𝑢)) = 𝑁)
24 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = ⟨𝑓, 𝑝⟩ → (2nd𝑢) = (2nd ‘⟨𝑓, 𝑝⟩))
259, 10op2nd 7068 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘⟨𝑓, 𝑝⟩) = 𝑝
2624, 25syl6req 2661 . . . . . . . . . . . . . . . . . . 19 (𝑢 = ⟨𝑓, 𝑝⟩ → 𝑝 = (2nd𝑢))
2726adantl 481 . . . . . . . . . . . . . . . . . 18 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → 𝑝 = (2nd𝑢))
2827adantr 480 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → 𝑝 = (2nd𝑢))
2918, 23, 28jca31 555 . . . . . . . . . . . . . . . 16 (((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3029ex 449 . . . . . . . . . . . . . . 15 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ 𝑢 = ⟨𝑓, 𝑝⟩) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3115, 30spcimedv 3265 . . . . . . . . . . . . . 14 (⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) → ((#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁 → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3231imp 444 . . . . . . . . . . . . 13 ((⟨𝑓, 𝑝⟩ ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘⟨𝑓, 𝑝⟩)) = 𝑁) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
338, 14, 32syl2anb 495 . . . . . . . . . . . 12 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
3433exlimiv 1845 . . . . . . . . . . 11 (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 𝑁) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
357, 34syl6bir 243 . . . . . . . . . 10 (𝑉 USGrph 𝐸 → (𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢))))
3635imp 444 . . . . . . . . 9 ((𝑉 USGrph 𝐸𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
37 fveq2 6103 . . . . . . . . . . . . . 14 (𝑝 = 𝑢 → (1st𝑝) = (1st𝑢))
3837fveq2d 6107 . . . . . . . . . . . . 13 (𝑝 = 𝑢 → (#‘(1st𝑝)) = (#‘(1st𝑢)))
3938eqeq1d 2612 . . . . . . . . . . . 12 (𝑝 = 𝑢 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑢)) = 𝑁))
4039elrab 3331 . . . . . . . . . . 11 (𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ↔ (𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁))
4140anbi1i 727 . . . . . . . . . 10 ((𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4241exbii 1764 . . . . . . . . 9 (∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)) ↔ ∃𝑢((𝑢 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑢)) = 𝑁) ∧ 𝑝 = (2nd𝑢)))
4336, 42sylibr 223 . . . . . . . 8 ((𝑉 USGrph 𝐸𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)))
44 df-rex 2902 . . . . . . . 8 (∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢) ↔ ∃𝑢(𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ∧ 𝑝 = (2nd𝑢)))
4543, 44sylibr 223 . . . . . . 7 ((𝑉 USGrph 𝐸𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢))
461rexeqi 3120 . . . . . . 7 (∃𝑢𝑇 𝑝 = (2nd𝑢) ↔ ∃𝑢 ∈ {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}𝑝 = (2nd𝑢))
4745, 46sylibr 223 . . . . . 6 ((𝑉 USGrph 𝐸𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ∃𝑢𝑇 𝑝 = (2nd𝑢))
48 fveq2 6103 . . . . . . . . 9 (𝑡 = 𝑢 → (2nd𝑡) = (2nd𝑢))
49 fvex 6113 . . . . . . . . 9 (2nd𝑢) ∈ V
5048, 3, 49fvmpt 6191 . . . . . . . 8 (𝑢𝑇 → (𝐹𝑢) = (2nd𝑢))
5150eqeq2d 2620 . . . . . . 7 (𝑢𝑇 → (𝑝 = (𝐹𝑢) ↔ 𝑝 = (2nd𝑢)))
5251rexbiia 3022 . . . . . 6 (∃𝑢𝑇 𝑝 = (𝐹𝑢) ↔ ∃𝑢𝑇 𝑝 = (2nd𝑢))
5347, 52sylibr 223 . . . . 5 ((𝑉 USGrph 𝐸𝑝 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
546, 53sylan2b 491 . . . 4 ((𝑉 USGrph 𝐸𝑝𝑊) → ∃𝑢𝑇 𝑝 = (𝐹𝑢))
5554ralrimiva 2949 . . 3 (𝑉 USGrph 𝐸 → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
5655adantr 480 . 2 ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢))
57 dffo3 6282 . 2 (𝐹:𝑇onto𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑝𝑊𝑢𝑇 𝑝 = (𝐹𝑢)))
585, 56, 57sylanbrc 695 1 ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℕ0cn0 11169  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   WWalksN cwwlkn 26206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wlknwwlknbij  26241
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