Theorem 2wspmdisj 30412

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 867	. . . . 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 30408	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
6	5	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
7	6	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
8	3, 4	fusgreg2wsplem 30408	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥)))
9		eqtr2 2757	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
10	9	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡‘1) = 𝑦 → ((𝑡‘1) = 𝑥 → 𝑥 = 𝑦))
11	10	adantl 481	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
14	8, 13	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑉 → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
15	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)))
16	15	imp 406	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
17	7, 16	sylbid 240	. . . . . . . . . 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 451	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑡 ∈ (𝑀‘𝑥) → ¬ 𝑡 ∈ (𝑀‘𝑦)))
20	19	ralrimiv 3127	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
21		disj 4402	. . . . . . 7 ⊢ (((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
22	20, 21	sylibr 234	. . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
23	22	olcd 874	. . . . 5 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
24	23	expcom 413	. . . 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 3176	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
27		fveq2 6834	. . 3 ⊢ (𝑥 = 𝑦 → (𝑀‘𝑥) = (𝑀‘𝑦))
28	27	disjor 5080	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
29	26, 28	mpbir 231	1 ⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399   ∩ cin 3900  ∅c0 4285  Disj wdisj 5065   ↦ cmpt 5179  ‘cfv 6492  (class class class)co 7358  1c1 11027  2c2 12200  Vtxcvtx 29069   WSPathsN cwwspthsn 29901

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-disj 5066  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361

This theorem is referenced by:  fusgreghash2wsp  30413