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Theorem 2wspmdisj 27512
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 399 . . . . 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 27508 . . . . . . . . . . . . 13 (𝑦𝑉 → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
65adantl 473 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
76adantr 472 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
83, 4fusgreg2wsplem 27508 . . . . . . . . . . . . . 14 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9 eqtr2 2780 . . . . . . . . . . . . . . . . . 18 (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
109expcom 450 . . . . . . . . . . . . . . . . 17 ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥𝑥 = 𝑦))
1110adantl 473 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 14 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
148, 13syl6bi 243 . . . . . . . . . . . . 13 (𝑥𝑉 → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1514adantr 472 . . . . . . . . . . . 12 ((𝑥𝑉𝑦𝑉) → (𝑡 ∈ (𝑀𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
1615imp 444 . . . . . . . . . . 11 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
177, 16sylbid 230 . . . . . . . . . 10 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (𝑡 ∈ (𝑀𝑦) → 𝑥 = 𝑦))
1817con3d 148 . . . . . . . . 9 (((𝑥𝑉𝑦𝑉) ∧ 𝑡 ∈ (𝑀𝑥)) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀𝑦)))
1918impancom 455 . . . . . . . 8 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀𝑥) → ¬ 𝑡 ∈ (𝑀𝑦)))
2019ralrimiv 3103 . . . . . . 7 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
21 disj 4160 . . . . . . 7 (((𝑀𝑥) ∩ (𝑀𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀𝑥) ¬ 𝑡 ∈ (𝑀𝑦))
2220, 21sylibr 224 . . . . . 6 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
2322olcd 407 . . . . 5 (((𝑥𝑉𝑦𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2423expcom 450 . . . 4 𝑥 = 𝑦 → ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)))
252, 24pm2.61i 176 . . 3 ((𝑥𝑉𝑦𝑉) → (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2625rgen2a 3115 . 2 𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅)
27 fveq2 6353 . . 3 (𝑥 = 𝑦 → (𝑀𝑥) = (𝑀𝑦))
2827disjor 4786 . 2 (Disj 𝑥𝑉 (𝑀𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥 = 𝑦 ∨ ((𝑀𝑥) ∩ (𝑀𝑦)) = ∅))
2926, 28mpbir 221 1 Disj 𝑥𝑉 (𝑀𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  cin 3714  c0 4058  Disj wdisj 4772  cmpt 4881  cfv 6049  (class class class)co 6814  1c1 10149  2c2 11282  Vtxcvtx 26094   WSPathsN cwwspthsn 26952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817
This theorem is referenced by:  fusgreghash2wsp  27513
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