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Theorem el2wlksoton 26405
 Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
Assertion
Ref Expression
el2wlksoton ((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
Distinct variable groups:   𝑇,𝑎,𝑏   𝐸,𝑎,𝑏   𝑉,𝑎,𝑏
Allowed substitution hints:   𝑋(𝑎,𝑏)   𝑌(𝑎,𝑏)

Proof of Theorem el2wlksoton
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 2wlksot 26394 . . 3 ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})
21eleq2d 2673 . 2 ((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ 𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)}))
3 eleq1 2676 . . . . 5 (𝑡 = 𝑇 → (𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
432rexbidv 3039 . . . 4 (𝑡 = 𝑇 → (∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) ↔ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
54elrab 3331 . . 3 (𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
65a1i 11 . 2 ((𝑉𝑋𝐸𝑌) → (𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)} ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))))
7 simpr 476 . . 3 ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)) → ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))
8 simpr 476 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)) → ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))
9 2wlkonot3v 26402 . . . . . . . 8 (𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑎𝑉𝑏𝑉) ∧ 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
109simp3d 1068 . . . . . . 7 (𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
1110a1i 11 . . . . . 6 ((𝑎𝑉𝑏𝑉) → (𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
1211rexlimivv 3018 . . . . 5 (∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
138, 12jccil 561 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
1413ex 449 . . 3 ((𝑉𝑋𝐸𝑌) → (∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏))))
157, 14impbid2 215 . 2 ((𝑉𝑋𝐸𝑌) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)) ↔ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
162, 6, 153bitrd 293 1 ((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   × cxp 5036  (class class class)co 6549   2WalksOt c2wlkot 26381   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 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-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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-2wlkonot 26385  df-2wlksot 26386 This theorem is referenced by:  el2wlksot  26407  2wot2wont  26413
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