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Theorem 2wspmdisj 28030
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 863 . . . . 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 28026 . . . . . . . . . . . . 13 (𝑦𝑉 → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
65adantl 482 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
76adantr 481 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
83, 4fusgreg2wsplem 28026 . . . . . . . . . . . . . 14 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9 eqtr2 2847 . . . . . . . . . . . . . . . . . 18 (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
109expcom 414 . . . . . . . . . . . . . . . . 17 ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥𝑥 = 𝑦))
1110adantl 482 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 14 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
148, 13syl6bi 254 . . . . . . . . . . . . 13 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1514adantr 481 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1615imp 407 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
177, 16sylbid 241 . . . . . . . . . 10 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) → 𝑥 = 𝑦))
1817con3d 155 . . . . . . . . 9 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀𝑦)))
1918impancom 452 . . . . . . . 8 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀𝑥) → ¬ 𝑡 ∈ (𝑀𝑦)))
2019ralrimiv 3186 . . . . . . 7 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
21 disj 4402 . . . . . . 7 (((𝑀𝑥) ∩ (𝑀𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
2220, 21sylibr 235 . . . . . 6 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
2322olcd 872 . . . . 5 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2423expcom 414 . . . 4 𝑥 = 𝑦 → ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)))
252, 24pm2.61i 183 . . 3 ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2625rgen2a 3234 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
27 fveq2 6667 . . 3 (𝑥 = 𝑦 → (𝑀𝑥) = (𝑀𝑦))
2827disjor 5043 . 2 (Disj 𝑥𝑉 (𝑀𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2926, 28mpbir 232 1 Disj 𝑥𝑉 (𝑀𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  wral 3143  {crab 3147  cin 3939  c0 4295  Disj wdisj 5028  cmpt 5143  cfv 6352  (class class class)co 7148  1c1 10527  2c2 11681  Vtxcvtx 26695   WSPathsN cwwspthsn 27520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-ov 7151
This theorem is referenced by:  fusgreghash2wsp  28031
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