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Theorem wlkwwlkinj 26850
Description: Lemma 2 for wlkwwlkbij2 26853. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-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
wlkwwlkinj ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)
Distinct variable groups:   𝐺,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇,𝑤   𝑡,𝑉   𝑡,𝑊   𝑤,𝐹   𝑤,𝑉
Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑡,𝑝)   𝑉(𝑝)   𝑊(𝑤,𝑝)

Proof of Theorem wlkwwlkinj
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 uspgrupgr 26116 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2 wlkwwlkbij.t . . . 4 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
3 wlkwwlkbij.w . . . 4 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
4 wlkwwlkbij.f . . . 4 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
52, 3, 4wlkwwlkfun 26849 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5syl3an1 1399 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
7 fveq2 6229 . . . . . . 7 (𝑡 = 𝑣 → (2nd𝑡) = (2nd𝑣))
8 fvex 6239 . . . . . . 7 (2nd𝑣) ∈ V
97, 4, 8fvmpt 6321 . . . . . 6 (𝑣𝑇 → (𝐹𝑣) = (2nd𝑣))
10 fveq2 6229 . . . . . . 7 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
11 fvex 6239 . . . . . . 7 (2nd𝑤) ∈ V
1210, 4, 11fvmpt 6321 . . . . . 6 (𝑤𝑇 → (𝐹𝑤) = (2nd𝑤))
139, 12eqeqan12d 2667 . . . . 5 ((𝑣𝑇𝑤𝑇) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
1413adantl 481 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
15 fveq2 6229 . . . . . . . . . 10 (𝑝 = 𝑣 → (1st𝑝) = (1st𝑣))
1615fveq2d 6233 . . . . . . . . 9 (𝑝 = 𝑣 → (#‘(1st𝑝)) = (#‘(1st𝑣)))
1716eqeq1d 2653 . . . . . . . 8 (𝑝 = 𝑣 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑣)) = 𝑁))
18 fveq2 6229 . . . . . . . . . 10 (𝑝 = 𝑣 → (2nd𝑝) = (2nd𝑣))
1918fveq1d 6231 . . . . . . . . 9 (𝑝 = 𝑣 → ((2nd𝑝)‘0) = ((2nd𝑣)‘0))
2019eqeq1d 2653 . . . . . . . 8 (𝑝 = 𝑣 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑣)‘0) = 𝑃))
2117, 20anbi12d 747 . . . . . . 7 (𝑝 = 𝑣 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
2221, 2elrab2 3399 . . . . . 6 (𝑣𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
23 fveq2 6229 . . . . . . . . . 10 (𝑝 = 𝑤 → (1st𝑝) = (1st𝑤))
2423fveq2d 6233 . . . . . . . . 9 (𝑝 = 𝑤 → (#‘(1st𝑝)) = (#‘(1st𝑤)))
2524eqeq1d 2653 . . . . . . . 8 (𝑝 = 𝑤 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑤)) = 𝑁))
26 fveq2 6229 . . . . . . . . . 10 (𝑝 = 𝑤 → (2nd𝑝) = (2nd𝑤))
2726fveq1d 6231 . . . . . . . . 9 (𝑝 = 𝑤 → ((2nd𝑝)‘0) = ((2nd𝑤)‘0))
2827eqeq1d 2653 . . . . . . . 8 (𝑝 = 𝑤 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑤)‘0) = 𝑃))
2925, 28anbi12d 747 . . . . . . 7 (𝑝 = 𝑤 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3029, 2elrab2 3399 . . . . . 6 (𝑤𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3122, 30anbi12i 733 . . . . 5 ((𝑣𝑇𝑤𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))))
32 3simpb 1079 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
3332adantr 480 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
34 simpl 472 . . . . . . . . 9 (((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃) → (#‘(1st𝑣)) = 𝑁)
3534anim2i 592 . . . . . . . 8 ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
3635adantr 480 . . . . . . 7 (((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
3736adantl 481 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
38 simpl 472 . . . . . . . . 9 (((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃) → (#‘(1st𝑤)) = 𝑁)
3938anim2i 592 . . . . . . . 8 ((𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
4039adantl 481 . . . . . . 7 (((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
4140adantl 481 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
42 uspgr2wlkeq2 26599 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4333, 37, 41, 42syl3anc 1366 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4431, 43sylan2b 491 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4514, 44sylbid 230 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
4645ralrimivva 3000 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
47 dff13 6552 . 2 (𝐹:𝑇1-1𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
486, 46, 47sylanbrc 699 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cmpt 4762  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  0cc0 9974  0cn0 11330  #chash 13157  UPGraphcupgr 26020  USPGraphcuspgr 26088  Walkscwlks 26548   WWalksN cwwlksn 26774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551  df-wwlks 26778  df-wwlksn 26779
This theorem is referenced by:  wlkwwlkbij  26852
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