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Theorem 2wspmdisj 28274
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 866 . . . . 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 28270 . . . . . . . . . . . . 13 (𝑦𝑉 → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
65adantl 485 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
76adantr 484 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
83, 4fusgreg2wsplem 28270 . . . . . . . . . . . . . 14 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9 eqtr2 2759 . . . . . . . . . . . . . . . . . 18 (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
109expcom 417 . . . . . . . . . . . . . . . . 17 ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥𝑥 = 𝑦))
1110adantl 485 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
148, 13syl6bi 256 . . . . . . . . . . . . 13 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1514adantr 484 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1615imp 410 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
177, 16sylbid 243 . . . . . . . . . 10 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) → 𝑥 = 𝑦))
1817con3d 155 . . . . . . . . 9 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀𝑦)))
1918impancom 455 . . . . . . . 8 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀𝑥) → ¬ 𝑡 ∈ (𝑀𝑦)))
2019ralrimiv 3095 . . . . . . 7 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
21 disj 4337 . . . . . . 7 (((𝑀𝑥) ∩ (𝑀𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
2220, 21sylibr 237 . . . . . 6 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
2322olcd 873 . . . . 5 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2423expcom 417 . . . 4 𝑥 = 𝑦 → ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)))
252, 24pm2.61i 185 . . 3 ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2625rgen2 3115 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
27 fveq2 6674 . . 3 (𝑥 = 𝑦 → (𝑀𝑥) = (𝑀𝑦))
2827disjor 5010 . 2 (Disj 𝑥𝑉 (𝑀𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2926, 28mpbir 234 1 Disj 𝑥𝑉 (𝑀𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wcel 2114  wral 3053  {crab 3057  cin 3842  c0 4211  Disj wdisj 4995  cmpt 5110  cfv 6339  (class class class)co 7170  1c1 10616  2c2 11771  Vtxcvtx 26941   WSPathsN cwwspthsn 27766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173
This theorem is referenced by:  fusgreghash2wsp  28275
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