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Theorem 2wspmdisj 30366
Description: The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.)
Hypotheses
Ref Expression
frgrhash2wsp.v 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
Assertion
Ref Expression
2wspmdisj Disj 𝑥𝑉 (𝑀𝑥)
Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑤,𝐺,𝑎,𝑥   𝑥,𝑉,𝑎,𝑤   𝑥,𝑀   𝑤,𝑉
Allowed substitution hints:   𝑀(𝑤,𝑎)

Proof of Theorem 2wspmdisj
Dummy variables 𝑦 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 867 . . . . 5 (𝑥 = 𝑦 → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
21a1d 25 . . . 4 (𝑥 = 𝑦 → ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)))
3 frgrhash2wsp.v . . . . . . . . . . . . . 14 𝑉 = (Vtx‘𝐺)
4 fusgreg2wsp.m . . . . . . . . . . . . . 14 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
53, 4fusgreg2wsplem 30362 . . . . . . . . . . . . 13 (𝑦𝑉 → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
65adantl 481 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
76adantr 480 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
83, 4fusgreg2wsplem 30362 . . . . . . . . . . . . . 14 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9 eqtr2 2759 . . . . . . . . . . . . . . . . . 18 (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
109expcom 413 . . . . . . . . . . . . . . . . 17 ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥𝑥 = 𝑦))
1110adantl 481 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → ((𝑡‘1) = 𝑥𝑥 = 𝑦))
1211com12 32 . . . . . . . . . . . . . . 15 ((𝑡‘1) = 𝑥 → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
1312adantl 481 . . . . . . . . . . . . . 14 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
148, 13biimtrdi 253 . . . . . . . . . . . . 13 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1514adantr 480 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1615imp 406 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
177, 16sylbid 240 . . . . . . . . . 10 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) → 𝑥 = 𝑦))
1817con3d 152 . . . . . . . . 9 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀𝑦)))
1918impancom 451 . . . . . . . 8 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀𝑥) → ¬ 𝑡 ∈ (𝑀𝑦)))
2019ralrimiv 3143 . . . . . . 7 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
21 disj 4456 . . . . . . 7 (((𝑀𝑥) ∩ (𝑀𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
2220, 21sylibr 234 . . . . . 6 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
2322olcd 874 . . . . 5 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2423expcom 413 . . . 4 𝑥 = 𝑦 → ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)))
252, 24pm2.61i 182 . . 3 ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2625rgen2 3197 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
27 fveq2 6907 . . 3 (𝑥 = 𝑦 → (𝑀𝑥) = (𝑀𝑦))
2827disjor 5130 . 2 (Disj 𝑥𝑉 (𝑀𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2926, 28mpbir 231 1 Disj 𝑥𝑉 (𝑀𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cin 3962  c0 4339  Disj wdisj 5115  cmpt 5231  cfv 6563  (class class class)co 7431  1c1 11154  2c2 12319  Vtxcvtx 29028   WSPathsN cwwspthsn 29858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434
This theorem is referenced by:  fusgreghash2wsp  30367
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