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Theorem wlkwwlkinj 27005
 Description: Lemma 2 for wlkwwlkbij2 27008. (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 26270 . . 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 27004 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5syl3an1 1167 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
7 fveq2 6352 . . . . . . 7 (𝑡 = 𝑣 → (2nd𝑡) = (2nd𝑣))
8 fvex 6362 . . . . . . 7 (2nd𝑣) ∈ V
97, 4, 8fvmpt 6444 . . . . . 6 (𝑣𝑇 → (𝐹𝑣) = (2nd𝑣))
10 fveq2 6352 . . . . . . 7 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
11 fvex 6362 . . . . . . 7 (2nd𝑤) ∈ V
1210, 4, 11fvmpt 6444 . . . . . 6 (𝑤𝑇 → (𝐹𝑤) = (2nd𝑤))
139, 12eqeqan12d 2776 . . . . 5 ((𝑣𝑇𝑤𝑇) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
1413adantl 473 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
15 fveq2 6352 . . . . . . . . . 10 (𝑝 = 𝑣 → (1st𝑝) = (1st𝑣))
1615fveq2d 6356 . . . . . . . . 9 (𝑝 = 𝑣 → (♯‘(1st𝑝)) = (♯‘(1st𝑣)))
1716eqeq1d 2762 . . . . . . . 8 (𝑝 = 𝑣 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑣)) = 𝑁))
18 fveq2 6352 . . . . . . . . . 10 (𝑝 = 𝑣 → (2nd𝑝) = (2nd𝑣))
1918fveq1d 6354 . . . . . . . . 9 (𝑝 = 𝑣 → ((2nd𝑝)‘0) = ((2nd𝑣)‘0))
2019eqeq1d 2762 . . . . . . . 8 (𝑝 = 𝑣 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑣)‘0) = 𝑃))
2117, 20anbi12d 749 . . . . . . 7 (𝑝 = 𝑣 → (((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
2221, 2elrab2 3507 . . . . . 6 (𝑣𝑇 ↔ (𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
23 fveq2 6352 . . . . . . . . . 10 (𝑝 = 𝑤 → (1st𝑝) = (1st𝑤))
2423fveq2d 6356 . . . . . . . . 9 (𝑝 = 𝑤 → (♯‘(1st𝑝)) = (♯‘(1st𝑤)))
2524eqeq1d 2762 . . . . . . . 8 (𝑝 = 𝑤 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑤)) = 𝑁))
26 fveq2 6352 . . . . . . . . . 10 (𝑝 = 𝑤 → (2nd𝑝) = (2nd𝑤))
2726fveq1d 6354 . . . . . . . . 9 (𝑝 = 𝑤 → ((2nd𝑝)‘0) = ((2nd𝑤)‘0))
2827eqeq1d 2762 . . . . . . . 8 (𝑝 = 𝑤 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑤)‘0) = 𝑃))
2925, 28anbi12d 749 . . . . . . 7 (𝑝 = 𝑤 → (((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3029, 2elrab2 3507 . . . . . 6 (𝑤𝑇 ↔ (𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3122, 30anbi12i 735 . . . . 5 ((𝑣𝑇𝑤𝑇) ↔ ((𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))))
32 3simpb 1145 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
3332adantr 472 . . . . . 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 474 . . . . . . . . 9 (((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃) → (♯‘(1st𝑣)) = 𝑁)
3534anim2i 594 . . . . . . . 8 ((𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) → (𝑣 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑣)) = 𝑁))
3635adantr 472 . . . . . . 7 (((𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑣 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑣)) = 𝑁))
3736adantl 473 . . . . . 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 474 . . . . . . . . 9 (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃) → (♯‘(1st𝑤)) = 𝑁)
3938anim2i 594 . . . . . . . 8 ((𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)) → (𝑤 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑤)) = 𝑁))
4039adantl 473 . . . . . . 7 (((𝑣 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑤 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑤)) = 𝑁))
4140adantl 473 . . . . . 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 26753 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑣)) = 𝑁) ∧ (𝑤 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝑤)) = 𝑁)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4333, 37, 41, 42syl3anc 1477 . . . . 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 493 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4514, 44sylbid 230 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
4645ralrimivva 3109 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
47 dff13 6675 . 2 (𝐹:𝑇1-1𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
486, 46, 47sylanbrc 701 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054   ↦ cmpt 4881  ⟶wf 6045  –1-1→wf1 6046  ‘cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  0cc0 10128  ℕ0cn0 11484  ♯chash 13311  UPGraphcupgr 26174  USPGraphcuspgr 26242  Walkscwlks 26702   WWalksN cwwlksn 26929 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-edg 26139  df-uhgr 26152  df-upgr 26176  df-uspgr 26244  df-wlks 26705  df-wwlks 26933  df-wwlksn 26934 This theorem is referenced by:  wlkwwlkbij  27007
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