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

Theorem 2wspmdisj 29323
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 866 . . . . 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 29319 . . . . . . . . . . . . 13 (𝑊 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑊) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊)))
65adantl 483 . . . . . . . . . . . 12 ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑊) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊)))
76adantr 482 . . . . . . . . . . 11 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑊) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊)))
83, 4fusgreg2wsplem 29319 . . . . . . . . . . . . . 14 (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9 eqtr2 2761 . . . . . . . . . . . . . . . . . 18 (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊)
109expcom 415 . . . . . . . . . . . . . . . . 17 ((𝑡‘1) = 𝑊 → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑊))
1110adantl 483 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊))
148, 13syl6bi 253 . . . . . . . . . . . . 13 (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊)))
1514adantr 482 . . . . . . . . . . . 12 ((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑊) → 𝑥 = 𝑊)))
1615imp 408 . . . . . . . . . . 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 453 . . . . . . . 8 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → (𝑡 ∈ (𝑀‘𝑥) → ¬ 𝑡 ∈ (𝑀‘𝑊)))
2019ralrimiv 3143 . . . . . . 7 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑊))
21 disj 4412 . . . . . . 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 873 . . . . 5 (((𝑥 ∈ 𝑉 ∧ 𝑊 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑊) → (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅))
2423expcom 415 . . . 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 3195 . 2 ∀𝑥 ∈ 𝑉 ∀𝑊 ∈ 𝑉 (𝑥 = 𝑊 √ ((𝑀‘𝑥) ∩ (𝑀‘𝑊)) = ∅)
27 fveq2 6847 . . 3 (𝑥 = 𝑊 → (𝑀‘𝑥) = (𝑀‘𝑊))
2827disjor 5090 . 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 397   √ wo 846   = wceq 1542   ∈ wcel 2107  âˆ€wral 3065  {crab 3410   ∩ cin 3914  âˆ…c0 4287  Disj wdisj 5075   ↩ cmpt 5193  â€˜cfv 6501  (class class class)co 7362  1c1 11059  2c2 12215  Vtxcvtx 27989   WSPathsN cwwspthsn 28815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365
This theorem is referenced by:  fusgreghash2wsp  29324
  Copyright terms: Public domain W3C validator