 Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlkwwlkinj Structured version   Visualization version   GIF version

Theorem wlkwwlkinj 41093
 Description: Lemma 2 for wlkwwlkbij2 41096. (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 ∈ (1Walks‘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 40406 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2 wlkwwlkbij.t . . . 4 𝑇 = {𝑝 ∈ (1Walks‘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 41092 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
61, 5syl3an1 1351 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
7 fveq2 6103 . . . . . . 7 (𝑡 = 𝑣 → (2nd𝑡) = (2nd𝑣))
8 fvex 6113 . . . . . . 7 (2nd𝑣) ∈ V
97, 4, 8fvmpt 6191 . . . . . 6 (𝑣𝑇 → (𝐹𝑣) = (2nd𝑣))
10 fveq2 6103 . . . . . . 7 (𝑡 = 𝑤 → (2nd𝑡) = (2nd𝑤))
11 fvex 6113 . . . . . . 7 (2nd𝑤) ∈ V
1210, 4, 11fvmpt 6191 . . . . . 6 (𝑤𝑇 → (𝐹𝑤) = (2nd𝑤))
139, 12eqeqan12d 2626 . . . . 5 ((𝑣𝑇𝑤𝑇) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
1413adantl 481 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) ↔ (2nd𝑣) = (2nd𝑤)))
15 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑣 → (1st𝑝) = (1st𝑣))
1615fveq2d 6107 . . . . . . . . 9 (𝑝 = 𝑣 → (#‘(1st𝑝)) = (#‘(1st𝑣)))
1716eqeq1d 2612 . . . . . . . 8 (𝑝 = 𝑣 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑣)) = 𝑁))
18 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑣 → (2nd𝑝) = (2nd𝑣))
1918fveq1d 6105 . . . . . . . . 9 (𝑝 = 𝑣 → ((2nd𝑝)‘0) = ((2nd𝑣)‘0))
2019eqeq1d 2612 . . . . . . . 8 (𝑝 = 𝑣 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑣)‘0) = 𝑃))
2117, 20anbi12d 743 . . . . . . 7 (𝑝 = 𝑣 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
2221, 2elrab2 3333 . . . . . 6 (𝑣𝑇 ↔ (𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)))
23 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑤 → (1st𝑝) = (1st𝑤))
2423fveq2d 6107 . . . . . . . . 9 (𝑝 = 𝑤 → (#‘(1st𝑝)) = (#‘(1st𝑤)))
2524eqeq1d 2612 . . . . . . . 8 (𝑝 = 𝑤 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑤)) = 𝑁))
26 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑤 → (2nd𝑝) = (2nd𝑤))
2726fveq1d 6105 . . . . . . . . 9 (𝑝 = 𝑤 → ((2nd𝑝)‘0) = ((2nd𝑤)‘0))
2827eqeq1d 2612 . . . . . . . 8 (𝑝 = 𝑤 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑤)‘0) = 𝑃))
2925, 28anbi12d 743 . . . . . . 7 (𝑝 = 𝑤 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3029, 2elrab2 3333 . . . . . 6 (𝑤𝑇 ↔ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))
3122, 30anbi12i 729 . . . . 5 ((𝑣𝑇𝑤𝑇) ↔ ((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))))
32 3simpb 1052 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
3332adantr 480 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → (𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0))
34 simpl 472 . . . . . . . . 9 (((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃) → (#‘(1st𝑣)) = 𝑁)
3534anim2i 591 . . . . . . . 8 ((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) → (𝑣 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
3635adantr 480 . . . . . . 7 (((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑣 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
3736adantl 481 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → (𝑣 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁))
38 simpl 472 . . . . . . . . 9 (((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃) → (#‘(1st𝑤)) = 𝑁)
3938anim2i 591 . . . . . . . 8 ((𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)) → (𝑤 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
4039adantl 481 . . . . . . 7 (((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃))) → (𝑤 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
4140adantl 481 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)))) → (𝑤 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁))
42 uspgr2wlkeq2 40855 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝑣 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑣)) = 𝑁) ∧ (𝑤 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝑤)) = 𝑁)) → ((2nd𝑣) = (2nd𝑤) → 𝑣 = 𝑤))
4333, 37, 41, 42syl3anc 1318 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ ((𝑣 ∈ (1Walks‘𝐺) ∧ ((#‘(1st𝑣)) = 𝑁 ∧ ((2nd𝑣)‘0) = 𝑃)) ∧ (𝑤 ∈ (1Walks‘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 229 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑣𝑇𝑤𝑇)) → ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
4645ralrimivva 2954 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤))
47 dff13 6416 . 2 (𝐹:𝑇1-1𝑊 ↔ (𝐹:𝑇𝑊 ∧ ∀𝑣𝑇𝑤𝑇 ((𝐹𝑣) = (𝐹𝑤) → 𝑣 = 𝑤)))
486, 46, 47sylanbrc 695 1 ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ↦ cmpt 4643  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕ0cn0 11169  #chash 12979   UPGraph cupgr 25747   USPGraph cuspgr 40378  1Walksc1wlks 40796   WWalkSN cwwlksn 41029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  wlkwwlkbij  41095
 Copyright terms: Public domain W3C validator