Theorem 2wspmdisj 28122

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 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})

Assertion

Ref	Expression
2wspmdisj	⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)

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.

Step	Hyp	Ref	Expression
1		orc 864	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
2	1	a1d 25	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)))
3		frgrhash2wsp.v	. . . . . . . . . . . . . 14 ⊢ 𝑉 = (Vtx‘𝐺)
4		fusgreg2wsp.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
5	3, 4	fusgreg2wsplem 28118	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
6	5	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
7	6	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
8	3, 4	fusgreg2wsplem 28118	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9		eqtr2 2819	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
10	9	expcom 417	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑦))
11	10	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑦))
12	11	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((𝑡‘1) = 𝑥 → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
13	12	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
14	8, 13	syl6bi 256	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
15	14	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
16	15	imp 410	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
17	7, 16	sylbid 243	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑦) → 𝑥 = 𝑦))
18	17	con3d 155	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦)))
19	18	impancom 455	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀‘𝑥) → ¬ 𝑡 ∈ (𝑀‘𝑦)))
20	19	ralrimiv 3148	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
21		disj 4355	. . . . . . 7 ⊢ (((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
22	20, 21	sylibr 237	. . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
23	22	olcd 871	. . . . 5 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
24	23	expcom 417	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)))
25	2, 24	pm2.61i 185	. . 3 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
26	25	rgen2 3168	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
27		fveq2 6645	. . 3 ⊢ (𝑥 = 𝑦 → (𝑀‘𝑥) = (𝑀‘𝑦))
28	27	disjor 5010	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
29	26, 28	mpbir 234	1 ⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ∩ cin 3880  ∅c0 4243  Disj wdisj 4995   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  1c1 10527  2c2 11680  Vtxcvtx 26789   WSPathsN cwwspthsn 27614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138

This theorem is referenced by:  fusgreghash2wsp  28123