MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2wspmdisj Structured version   Visualization version   GIF version

Theorem 2wspmdisj 29854
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 = (𝑎 ∈ 𝑉 ↩ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑎})
Assertion
Ref Expression
2wspmdisj Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)
Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑀,𝐺,𝑎,𝑥   𝑥,𝑉,𝑎,𝑀   𝑥,𝑀   𝑀,𝑉
Allowed substitution hints:   𝑀(𝑀,𝑎)

Proof of Theorem 2wspmdisj
Dummy variables 𝑊 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 orc 864 . . . . 5 (𝑥 = 𝑊 → (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅))
21a1d 25 . . . 4 (𝑥 = 𝑊 → ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅)))
3 frgrhash2wsp.v . . . . . . . . . . . . . 14 𝑉 = (Vtx‘𝐺)
4 fusgreg2wsp.m . . . . . . . . . . . . . 14 𝑀 = (𝑎 ∈ 𝑉 ↩ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑎})
53, 4fusgreg2wsplem 29850 . . . . . . . . . . . . 13 (𝑊 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑊) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊)))
65adantl 481 . . . . . . . . . . . 12 ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑊) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊)))
76adantr 480 . . . . . . . . . . 11 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑊) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊)))
83, 4fusgreg2wsplem 29850 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9 eqtr2 2755 . . . . . . . . . . . . . . . . . 18 (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊)
109expcom 413 . . . . . . . . . . . . . . . . 17 ((𝑡‘1) = 𝑊 → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑊))
1110adantl 481 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑊))
1211com12 32 . . . . . . . . . . . . . . 15 ((𝑡‘1) = 𝑥 → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊))
1312adantl 481 . . . . . . . . . . . . . 14 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊))
148, 13syl6bi 252 . . . . . . . . . . . . 13 (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊)))
1514adantr 480 . . . . . . . . . . . 12 ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊)))
1615imp 406 . . . . . . . . . . 11 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊))
177, 16sylbid 239 . . . . . . . . . 10 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑊) → 𝑥 = 𝑊))
1817con3d 152 . . . . . . . . 9 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (¬ 𝑥 = 𝑊 → ¬ 𝑡 ∈ (𝑀‘𝑊)))
1918impancom 451 . . . . . . . 8 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → (𝑡 ∈ (𝑀‘𝑥) → ¬ 𝑡 ∈ (𝑀‘𝑊)))
2019ralrimiv 3144 . . . . . . 7 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑊))
21 disj 4448 . . . . . . 7 (((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅ ↔ ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑊))
2220, 21sylibr 233 . . . . . 6 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅)
2322olcd 871 . . . . 5 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅))
2423expcom 413 . . . 4 (¬ 𝑥 = 𝑊 → ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅)))
252, 24pm2.61i 182 . . 3 ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅))
2625rgen2 3196 . 2 ∀𝑥 ∈ 𝑉 ∀𝑊 ∈ 𝑉 (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅)
27 fveq2 6892 . . 3 (𝑥 = 𝑊 → (𝑀‘𝑥) = (𝑀‘𝑊))
2827disjor 5129 . 2 (Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑊 ∈ 𝑉 (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅))
2926, 28mpbir 230 1 Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)
Colors of variables: wff setvar class
Syntax hints:  Â¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   √ wo 844   = wceq 1540   ∈ wcel 2105  âˆ€wral 3060  {crab 3431   ∩ cin 3948  âˆ…c0 4323  Disj wdisj 5114   ↩ cmpt 5232  â€˜cfv 6544  (class class class)co 7412  1c1 11114  2c2 12272  Vtxcvtx 28520   WSPathsN cwwspthsn 29346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7415
This theorem is referenced by:  fusgreghash2wsp  29855
  Copyright terms: Public domain W3C validator