Theorem 2wspmdisj 41501

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.)

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   𝑥,𝐺,𝑤,𝑎   𝑥,𝑉,𝑤,𝑎   𝑥,𝑀

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 399	. . . . 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		fusgreg2wsp.m	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
4	3	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}))
5		eqeq2 2621	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑥 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑥))
6	5	rabbidv 3164	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑥 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
7	6	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑎 = 𝑥) → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
8		simpl 472	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
9		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (2 WSPathsN 𝐺) ∈ V
10	9	rabex 4740	. . . . . . . . . . . . . 14 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ∈ V
11	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ∈ V)
12	4, 7, 8, 11	fvmptd 6197	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑀‘𝑥) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥})
13	12	eleq2d 2673	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) ↔ 𝑡 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥}))
14		fveq1 6102	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑡 → (𝑤‘1) = (𝑡‘1))
15	14	eqeq1d 2612	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑡 → ((𝑤‘1) = 𝑥 ↔ (𝑡‘1) = 𝑥))
16	15	elrab 3331	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥))
17		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝑦 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑦))
18	17	rabbidv 3164	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑦 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦})
19	18	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ 𝑎 = 𝑦) → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦})
20		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
21	9	rabex 4740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦} ∈ V
22	21	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦} ∈ V)
23	4, 19, 20, 22	fvmptd 6197	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑀‘𝑦) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦})
24	23	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → (𝑀‘𝑦) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦})
25	24	eleq2d 2673	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → (𝑡 ∈ (𝑀‘𝑦) ↔ 𝑡 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦}))
26	14	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑡 → ((𝑤‘1) = 𝑦 ↔ (𝑡‘1) = 𝑦))
27	26	elrab 3331	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑦} ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦))
28	25, 27	syl6bb 275	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦)))
29		eqtr2 2630	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑡‘1) = 𝑥 ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦)
30	29	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡‘1) = 𝑥 → ((𝑡‘1) = 𝑦 → 𝑥 = 𝑦))
31	30	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → ((𝑡‘1) = 𝑦 → 𝑥 = 𝑦))
32	31	adantld 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑦) → 𝑥 = 𝑦))
33	28, 32	sylbid 229	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → (𝑡 ∈ (𝑀‘𝑦) → 𝑥 = 𝑦))
34	33	con3d 147	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝑡‘1) = 𝑥) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦)))
35	34	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑡‘1) = 𝑥 → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦))))
36	35	adantld 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑡 ∈ (2 WSPathsN 𝐺) ∧ (𝑡‘1) = 𝑥) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦))))
37	16, 36	syl5bi 231	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑥} → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦))))
38	13, 37	sylbid 229	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑡 ∈ (𝑀‘𝑥) → (¬ 𝑥 = 𝑦 → ¬ 𝑡 ∈ (𝑀‘𝑦))))
39	38	com23 84	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → (𝑡 ∈ (𝑀‘𝑥) → ¬ 𝑡 ∈ (𝑀‘𝑦))))
40	39	imp31 447	. . . . . . . 8 ⊢ ((((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ¬ 𝑡 ∈ (𝑀‘𝑦))
41	40	ralrimiva 2949	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
42		disj 3969	. . . . . . 7 ⊢ (((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
43	41, 42	sylibr 223	. . . . . 6 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
44	43	olcd 407	. . . . 5 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
45	44	expcom 450	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)))
46	2, 45	pm2.61i 175	. . 3 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
47	46	rgen2 2958	. 2 ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
48		fveq2 6103	. . 3 ⊢ (𝑥 = 𝑦 → (𝑀‘𝑥) = (𝑀‘𝑦))
49	48	disjor 4567	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
50	47, 49	mpbir 220	1 ⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∩ cin 3539  ∅c0 3874  Disj wdisj 4553   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1c1 9816  2c2 10947  Vtxcvtx 25673   WSPathsN cwwspthsn 41031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by:  fusgreghash2wsp  41502