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Theorem 2pthwlkonot 26412
 Description: For two different vertices, a walk of length 2 between these vertices as ordered triple is a simple path of length 2 between these vertices as ordered triple in an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
2pthwlkonot ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))

Proof of Theorem 2pthwlkonot
Dummy variables 𝑓 𝑝 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr1 1060 . . . . . . . . . . 11 (((#‘𝑓) = 2 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵)) → 𝑉 USGrph 𝐸)
2 simpl 472 . . . . . . . . . . 11 (((#‘𝑓) = 2 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵)) → (#‘𝑓) = 2)
3 simpr3 1062 . . . . . . . . . . 11 (((#‘𝑓) = 2 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵)) → 𝐴𝐵)
4 usgra2wlkspth 26149 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ (#‘𝑓) = 2 ∧ 𝐴𝐵) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝))
51, 2, 3, 4syl3anc 1318 . . . . . . . . . 10 (((#‘𝑓) = 2 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝))
65bicomd 212 . . . . . . . . 9 (((#‘𝑓) = 2 ∧ (𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝))
76ex 449 . . . . . . . 8 ((#‘𝑓) = 2 → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)))
87adantr 480 . . . . . . 7 (((#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)) → ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)))
98com12 32 . . . . . 6 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (((#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)))
109pm5.32rd 670 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ ((#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ ((#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)))))
11 3anass 1035 . . . . 5 ((𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ ((#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))))
12 3anass 1035 . . . . 5 ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ ((#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))))
1310, 11, 123bitr4g 302 . . . 4 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))))
14132exbidv 1839 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵)) ↔ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))))
1514rabbidv 3164 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
16 usgrav 25867 . . . 4 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
17 2spthonot 26393 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
1816, 17sylan 487 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
19183adant3 1074 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
20 2wlkonot 26392 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
2116, 20sylan 487 . . 3 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
22213adant3 1074 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})
2315, 19, 223eqtr4d 2654 1 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1c1 9816  2c2 10947  #chash 12979   USGrph cusg 25859   WalkOn cwlkon 26030   SPathOn cspthon 26033   2WalksOnOt c2wlkonot 26382   2SPathOnOt c2pthonot 26384 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-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-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-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2wlkonot 26385  df-2spthonot 26387 This theorem is referenced by:  usg2spthonot  26415  usg2spthonot0  26416  frg2spot1  26585
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