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Theorem el2wlkonotot1 26401
 Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
Assertion
Ref Expression
el2wlkonotot1 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))

Proof of Theorem el2wlkonotot1
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el2wlkonotot0 26399 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
2 simpll 786 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (𝑉𝑋𝐸𝑌))
3 eleq1 2676 . . . . . . . . . . 11 (𝑅 = 𝐴 → (𝑅𝑉𝐴𝑉))
43eqcoms 2618 . . . . . . . . . 10 (𝐴 = 𝑅 → (𝑅𝑉𝐴𝑉))
5 eleq1 2676 . . . . . . . . . . 11 (𝑆 = 𝐶 → (𝑆𝑉𝐶𝑉))
65eqcoms 2618 . . . . . . . . . 10 (𝐶 = 𝑆 → (𝑆𝑉𝐶𝑉))
74, 6bi2anan9 913 . . . . . . . . 9 ((𝐴 = 𝑅𝐶 = 𝑆) → ((𝑅𝑉𝑆𝑉) ↔ (𝐴𝑉𝐶𝑉)))
87biimpcd 238 . . . . . . . 8 ((𝑅𝑉𝑆𝑉) → ((𝐴 = 𝑅𝐶 = 𝑆) → (𝐴𝑉𝐶𝑉)))
98adantl 481 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → ((𝐴 = 𝑅𝐶 = 𝑆) → (𝐴𝑉𝐶𝑉)))
109imp 444 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (𝐴𝑉𝐶𝑉))
11 el2wlkonotot 26400 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
122, 10, 11syl2anc 691 . . . . 5 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
1312bicomd 212 . . . 4 ((((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) ∧ (𝐴 = 𝑅𝐶 = 𝑆)) → (∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
1413pm5.32da 671 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (((𝐴 = 𝑅𝐶 = 𝑆) ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅𝐶 = 𝑆) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
15 df-3an 1033 . . 3 ((𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅𝐶 = 𝑆) ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
16 df-3an 1033 . . 3 ((𝐴 = 𝑅𝐶 = 𝑆 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)) ↔ ((𝐴 = 𝑅𝐶 = 𝑆) ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶)))
1714, 15, 163bitr4g 302 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → ((𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
181, 17bitrd 267 1 (((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cotp 4133   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382 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-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-ot 4134  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-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-wlk 26036  df-wlkon 26042  df-2wlkonot 26385 This theorem is referenced by:  usg2spthonot0  26416
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