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Theorem wlkwwlkinj 26645
 Description: Lemma 2 for wlkwwlkbij2 26648. (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 25958 . . 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 26644 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5syl3an1 1356 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
7 fveq2 6150 . . . . . . 7 (𝑡 = 𝑣 → (2nd𝑡) = (2nd𝑣))
8 fvex 6160 . . . . . . 7 (2nd𝑣) ∈ V
97, 4, 8fvmpt 6240 . . . . . 6 (𝑣𝑇 → (𝐹𝑣) = (2nd𝑣))
10 fveq2 6150 . . . . . . 7 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
11 fvex 6160 . . . . . . 7 (2nd𝑤) ∈ V
1210, 4, 11fvmpt 6240 . . . . . 6 (𝑤𝑇 → (𝐹𝑤) = (2nd𝑤))
139, 12eqeqan12d 2642 . . . . 5 ((𝑣𝑇𝑤𝑇) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
1413adantl 482 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
15 fveq2 6150 . . . . . . . . . 10 (𝑝 = 𝑣 → (1st𝑝) = (1st𝑣))
1615fveq2d 6154 . . . . . . . . 9 (𝑝 = 𝑣 → (#‘(1st𝑝)) = (#‘(1st𝑣)))
1716eqeq1d 2628 . . . . . . . 8 (𝑝 = 𝑣 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑣)) = 𝑁))
18 fveq2 6150 . . . . . . . . . 10 (𝑝 = 𝑣 → (2nd𝑝) = (2nd𝑣))
1918fveq1d 6152 . . . . . . . . 9 (𝑝 = 𝑣 → ((2nd𝑝)‘0) = ((2nd𝑣)‘0))
2019eqeq1d 2628 . . . . . . . 8 (𝑝 = 𝑣 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑣)‘0) = 𝑃))
2117, 20anbi12d 746 . . . . . . 7 (𝑝 = 𝑣 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
2221, 2elrab2 3353 . . . . . 6 (𝑣𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
23 fveq2 6150 . . . . . . . . . 10 (𝑝 = 𝑤 → (1st𝑝) = (1st𝑤))
2423fveq2d 6154 . . . . . . . . 9 (𝑝 = 𝑤 → (#‘(1st𝑝)) = (#‘(1st𝑤)))
2524eqeq1d 2628 . . . . . . . 8 (𝑝 = 𝑤 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑤)) = 𝑁))
26 fveq2 6150 . . . . . . . . . 10 (𝑝 = 𝑤 → (2nd𝑝) = (2nd𝑤))
2726fveq1d 6152 . . . . . . . . 9 (𝑝 = 𝑤 → ((2nd𝑝)‘0) = ((2nd𝑤)‘0))
2827eqeq1d 2628 . . . . . . . 8 (𝑝 = 𝑤 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑤)‘0) = 𝑃))
2925, 28anbi12d 746 . . . . . . 7 (𝑝 = 𝑤 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3029, 2elrab2 3353 . . . . . 6 (𝑤𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3122, 30anbi12i 732 . . . . 5 ((𝑣𝑇𝑤𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))))
32 3simpb 1057 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
3332adantr 481 . . . . . 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 473 . . . . . . . . 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 481 . . . . . . 7 (((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
3736adantl 482 . . . . . 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 473 . . . . . . . . 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 482 . . . . . . 7 (((𝑣 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
4140adantl 482 . . . . . 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 26406 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4333, 37, 41, 42syl3anc 1323 . . . . 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 492 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4514, 44sylbid 230 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
4645ralrimivva 2970 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
47 dff13 6467 . 2 (𝐹:𝑇1-1𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
486, 46, 47sylanbrc 697 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992  ∀wral 2912  {crab 2916   ↦ cmpt 4678  ⟶wf 5846  –1-1→wf1 5847  ‘cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  0cc0 9881  ℕ0cn0 11237  #chash 13054   UPGraph cupgr 25866   USPGraph cuspgr 25931  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-uspgr 25933  df-wlks 26359  df-wwlks 26585  df-wwlksn 26586 This theorem is referenced by:  wlkwwlkbij  26647
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