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Theorem wlkwwlkfun 26650
Description: Lemma 1 for wlkwwlkbij2 26654. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
Hypotheses
Ref Expression
wlkwwlkbij.t 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
wlkwwlkbij.w 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
wlkwwlkbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlkwwlkfun ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
Distinct variable groups:   𝐺,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇,𝑤   𝑡,𝑉   𝑡,𝑊
Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑤,𝑡,𝑝)   𝑉(𝑤,𝑝)   𝑊(𝑤,𝑝)

Proof of Theorem wlkwwlkfun
StepHypRef Expression
1 fveq2 6148 . . . . . . . 8 (𝑝 = 𝑡 → (1st𝑝) = (1st𝑡))
21fveq2d 6152 . . . . . . 7 (𝑝 = 𝑡 → (#‘(1st𝑝)) = (#‘(1st𝑡)))
32eqeq1d 2623 . . . . . 6 (𝑝 = 𝑡 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑡)) = 𝑁))
4 fveq2 6148 . . . . . . . 8 (𝑝 = 𝑡 → (2nd𝑝) = (2nd𝑡))
54fveq1d 6150 . . . . . . 7 (𝑝 = 𝑡 → ((2nd𝑝)‘0) = ((2nd𝑡)‘0))
65eqeq1d 2623 . . . . . 6 (𝑝 = 𝑡 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑡)‘0) = 𝑃))
73, 6anbi12d 746 . . . . 5 (𝑝 = 𝑡 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)))
8 wlkwwlkbij.t . . . . 5 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
97, 8elrab2 3348 . . . 4 (𝑡𝑇 ↔ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)))
10 simp1 1059 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐺 ∈ UPGraph )
11 simpl 473 . . . . . . 7 ((𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)) → 𝑡 ∈ (Walks‘𝐺))
1210, 11anim12i 589 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)))
13 simp3 1061 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
14 simprl 793 . . . . . . 7 ((𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)) → (#‘(1st𝑡)) = 𝑁)
1513, 14anim12i 589 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑡)) = 𝑁))
16 wlknewwlksn 26642 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑡)) = 𝑁)) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
1712, 15, 16syl2anc 692 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
18 simprrr 804 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → ((2nd𝑡)‘0) = 𝑃)
1917, 18jca 554 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → ((2nd𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd𝑡)‘0) = 𝑃))
209, 19sylan2b 492 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → ((2nd𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd𝑡)‘0) = 𝑃))
21 fveq1 6147 . . . . 5 (𝑤 = (2nd𝑡) → (𝑤‘0) = ((2nd𝑡)‘0))
2221eqeq1d 2623 . . . 4 (𝑤 = (2nd𝑡) → ((𝑤‘0) = 𝑃 ↔ ((2nd𝑡)‘0) = 𝑃))
23 wlkwwlkbij.w . . . 4 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
2422, 23elrab2 3348 . . 3 ((2nd𝑡) ∈ 𝑊 ↔ ((2nd𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd𝑡)‘0) = 𝑃))
2520, 24sylibr 224 . 2 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → (2nd𝑡) ∈ 𝑊)
26 wlkwwlkbij.f . 2 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
2725, 26fmptd 6340 1 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880  0cn0 11236  #chash 13057   UPGraph cupgr 25871  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-wlks 26365  df-wwlks 26591  df-wwlksn 26592
This theorem is referenced by:  wlkwwlkinj  26651  wlkwwlksur  26652
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