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Theorem 2wot2wont 26413
 Description: The set of (simple) paths of length 2 (in a graph) is the set of (simple) paths of length 2 between any two different vertices. (Contributed by Alexander van der Vekens, 27-Feb-2018.)
Assertion
Ref Expression
2wot2wont ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = 𝑥𝑉 𝑦𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))
Distinct variable groups:   𝑥,𝐸,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem 2wot2wont
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el2wlksoton 26405 . . . 4 ((𝑉𝑋𝐸𝑌) → (𝑤 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑥𝑉𝑦𝑉 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)))
2 vex 3176 . . . . 5 𝑤 ∈ V
3 eleq1 2676 . . . . . 6 (𝑢 = 𝑤 → (𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦) ↔ 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)))
432rexbidv 3039 . . . . 5 (𝑢 = 𝑤 → (∃𝑥𝑉𝑦𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦) ↔ ∃𝑥𝑉𝑦𝑉 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)))
52, 4elab 3319 . . . 4 (𝑤 ∈ {𝑢 ∣ ∃𝑥𝑉𝑦𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)} ↔ ∃𝑥𝑉𝑦𝑉 𝑤 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))
61, 5syl6bbr 277 . . 3 ((𝑉𝑋𝐸𝑌) → (𝑤 ∈ (𝑉 2WalksOt 𝐸) ↔ 𝑤 ∈ {𝑢 ∣ ∃𝑥𝑉𝑦𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)}))
76eqrdv 2608 . 2 ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = {𝑢 ∣ ∃𝑥𝑉𝑦𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)})
8 dfiunv2 4492 . 2 𝑥𝑉 𝑦𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦) = {𝑢 ∣ ∃𝑥𝑉𝑦𝑉 𝑢 ∈ (𝑥(𝑉 2WalksOnOt 𝐸)𝑦)}
97, 8syl6eqr 2662 1 ((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = 𝑥𝑉 𝑦𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  ∪ ciun 4455  (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: (None)
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