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Theorem wlknwwlksnfun 26637
 Description: Lemma 1 for wlknwwlksnbij2 26641. (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
wlknwwlksnfun ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
Distinct variable groups:   𝐺,𝑝,𝑡   𝑁,𝑝,𝑡   𝑡,𝑇   𝑡,𝑊
Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑡,𝑝)   𝑊(𝑝)

Proof of Theorem wlknwwlksnfun
StepHypRef Expression
1 fveq2 6150 . . . . . . 7 (𝑝 = 𝑡 → (1st𝑝) = (1st𝑡))
21fveq2d 6154 . . . . . 6 (𝑝 = 𝑡 → (#‘(1st𝑝)) = (#‘(1st𝑡)))
32eqeq1d 2628 . . . . 5 (𝑝 = 𝑡 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑡)) = 𝑁))
4 wlknwwlksnbij.t . . . . 5 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}
53, 4elrab2 3353 . . . 4 (𝑡𝑇 ↔ (𝑡 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑡)) = 𝑁))
6 wlknewwlksn 26636 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑡)) = 𝑁)) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
76an4s 868 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑡)) = 𝑁)) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
85, 7sylan2b 492 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
9 wlknwwlksnbij.w . . 3 𝑊 = (𝑁 WWalksN 𝐺)
108, 9syl6eleqr 2715 . 2 (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → (2nd𝑡) ∈ 𝑊)
11 wlknwwlksnbij.f . 2 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
1210, 11fmptd 6341 1 ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  {crab 2916   ↦ cmpt 4678  ⟶wf 5846  ‘cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  ℕ0cn0 11237  #chash 13054   UPGraph cupgr 25866  Walkscwlks 26356   WWalksN cwwlksn 26581 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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-edg 25835  df-uhgr 25844  df-upgr 25868  df-wlks 26359  df-wwlks 26585  df-wwlksn 26586 This theorem is referenced by:  wlknwwlksninj  26638  wlknwwlksnsur  26639
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