Theorem 2wspmdisj 29281

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 865	. . . . 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 29277	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
6	5	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
7	6	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
8	3, 4	fusgreg2wsplem 29277	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9		eqtr2 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
10	9	expcom 414	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑦))
11	10	adantl 482	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
14	8, 13	syl6bi 252	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
15	14	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
16	15	imp 407	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
17	7, 16	sylbid 239	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑦) → 𝑥 = 𝑦))
18	17	con3d 152	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦)))
19	18	impancom 452	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀‘𝑥) → ¬ 𝑡 ∈ (𝑀‘𝑦)))
20	19	ralrimiv 3142	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
21		disj 4407	. . . . . . 7 ⊢ (((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
22	20, 21	sylibr 233	. . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
23	22	olcd 872	. . . . 5 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
24	23	expcom 414	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)))
25	2, 24	pm2.61i 182	. . 3 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
26	25	rgen2 3194	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
27		fveq2 6842	. . 3 ⊢ (𝑥 = 𝑦 → (𝑀‘𝑥) = (𝑀‘𝑦))
28	27	disjor 5085	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
29	26, 28	mpbir 230	1 ⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407   ∩ cin 3909  ∅c0 4282  Disj wdisj 5070   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357  1c1 11052  2c2 12208  Vtxcvtx 27947   WSPathsN cwwspthsn 28773

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360

This theorem is referenced by:  fusgreghash2wsp  29282