MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkwwlkinj Structured version   Visualization version   GIF version

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  1-1wf1 6046  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  0cc0 10128  0cn0 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
  Copyright terms: Public domain W3C validator