Theorem 2spotmdisj 26595

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, 17-Sep-2021.)

Hypothesis

Ref	Expression
usgreghash2spot.m	⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)})

Assertion

Ref	Expression
2spotmdisj	⊢ (𝑉 ∈ 𝑊 → Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥))

Distinct variable groups:   𝑡,𝐸,𝑥,𝑎   𝑉,𝑎,𝑡,𝑥   𝐸,𝑎   𝑡,𝑀,𝑥   𝑡,𝑊,𝑥

Allowed substitution hints:   𝑀(𝑎)   𝑊(𝑎)

Proof of Theorem 2spotmdisj

Dummy variables 𝑦 𝑐 𝑑 𝑏 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 → ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)))
3		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉)
4		3xpexg 6859	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ 𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
5		rabexg 4739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V)
6	4, 5	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 ∈ 𝑊 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V)
7		eqeq2 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑥 → ((2nd ‘(1st ‘𝑡)) = 𝑎 ↔ (2nd ‘(1st ‘𝑡)) = 𝑥))
8	7	anbi2d 736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑥 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)))
9	8	rabbidv 3164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)})
10		usgreghash2spot.m	. . . . . . . . . . . . . . . . 17 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)})
11	9, 10	fvmptg 6189	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V) → (𝑀‘𝑥) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)})
12	3, 6, 11	syl2anr 494	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑀‘𝑥) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)})
13	12	eleq2d 2673	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑡 ∈ (𝑀‘𝑥) ↔ 𝑡 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)}))
14		rabid 3095	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ↔ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)))
15		el2xptp 7102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑡 = ⟨𝑏, 𝑐, 𝑑⟩)
16		ot2ndg 7074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝑑 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑐)
17	16	3expa 1257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑐)
18	17	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑥 ↔ 𝑐 = 𝑥))
19	17	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 = 𝑥) → (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑐)
20	19	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 = 𝑥) → ((2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦 ↔ 𝑐 = 𝑦))
21		ax7 1930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑐 = 𝑥 → (𝑐 = 𝑦 → 𝑥 = 𝑦))
22	21	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 = 𝑥) → (𝑐 = 𝑦 → 𝑥 = 𝑦))
23	20, 22	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 = 𝑥) → ((2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦 → 𝑥 = 𝑦))
24	23	con3d 147	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) ∧ 𝑐 = 𝑥) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))
25	24	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → (𝑐 = 𝑥 → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦)))
26	25	a1dd 48	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → (𝑐 = 𝑥 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))))
27	18, 26	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑥 → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))))
28	27	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑥 → (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))))
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑥 → (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦)))))
30		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → (1st ‘𝑡) = (1st ‘⟨𝑏, 𝑐, 𝑑⟩))
31	30	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)))
32	31	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((2nd ‘(1st ‘𝑡)) = 𝑥 ↔ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑥))
33	31	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((2nd ‘(1st ‘𝑡)) = 𝑦 ↔ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))
34	33	notbid 307	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → (¬ (2nd ‘(1st ‘𝑡)) = 𝑦 ↔ ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))
35	34	imbi2d 329	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦) ↔ (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦)))
36	35	imbi2d 329	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)) ↔ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦))))
37	36	imbi2d 329	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦))) ↔ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘⟨𝑏, 𝑐, 𝑑⟩)) = 𝑦)))))
38	29, 32, 37	3imtr4d 282	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((2nd ‘(1st ‘𝑡)) = 𝑥 → (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))))
39	38	com12 32	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((2nd ‘(1st ‘𝑡)) = 𝑥 → (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))))
40	39	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥) → (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))))
41	40	com13 86	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ 𝑑 ∈ 𝑉) → (𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))))
42	41	rexlimdva 3013	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (∃𝑑 ∈ 𝑉 𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))))
43	42	rexlimivv 3018	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 𝑡 = ⟨𝑏, 𝑐, 𝑑⟩ → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦))))
44	15, 43	sylbi 206	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦))))
45	44	imp 444	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))
46	45	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))
47	46	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → ((𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))
48	14, 47	syl5bi 231	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑡 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))
49	13, 48	sylbid 229	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑡 ∈ (𝑀‘𝑥) → (¬ 𝑥 = 𝑦 → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))
50	49	com23 84	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (¬ 𝑥 = 𝑦 → (𝑡 ∈ (𝑀‘𝑥) → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)))
51	50	imp31 447	. . . . . . . . . . 11 ⊢ ((((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ¬ (2nd ‘(1st ‘𝑡)) = 𝑦)
52	51	intnand 953	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ¬ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦))
53	52	intnand 953	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ¬ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)))
54		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
55		rabexg 4739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)} ∈ V)
56	4, 55	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝑊 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)} ∈ V)
57		eqeq2 2621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑦 → ((2nd ‘(1st ‘𝑡)) = 𝑎 ↔ (2nd ‘(1st ‘𝑡)) = 𝑦))
58	57	anbi2d 736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑦 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)))
59	58	rabbidv 3164	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑦 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)})
60	59, 10	fvmptg 6189	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑉 ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)} ∈ V) → (𝑀‘𝑦) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)})
61	54, 56, 60	syl2anr 494	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑀‘𝑦) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)})
62	61	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑡 ∈ (𝑀‘𝑦) ↔ 𝑡 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)}))
63		rabid 3095	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)} ↔ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦)))
64	62, 63	syl6bb 275	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑡 ∈ (𝑀‘𝑦) ↔ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦))))
65	64	notbid 307	. . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (¬ 𝑡 ∈ (𝑀‘𝑦) ↔ ¬ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦))))
66	65	adantr 480	. . . . . . . . . 10 ⊢ (((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) → (¬ 𝑡 ∈ (𝑀‘𝑦) ↔ ¬ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦))))
67	66	adantr 480	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) ∧ 𝑡 ∈ (𝑀‘𝑥)) → (¬ 𝑡 ∈ (𝑀‘𝑦) ↔ ¬ (𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑦))))
68	53, 67	mpbird 246	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) ∧ 𝑡 ∈ (𝑀‘𝑥)) → ¬ 𝑡 ∈ (𝑀‘𝑦))
69	68	ralrimiva 2949	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) → ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
70		disj 3969	. . . . . . 7 ⊢ (((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅ ↔ ∀𝑡 ∈ (𝑀‘𝑥) ¬ 𝑡 ∈ (𝑀‘𝑦))
71	69, 70	sylibr 223	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) → ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)
72	71	olcd 407	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) ∧ ¬ 𝑥 = 𝑦) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
73	72	expcom 450	. . . 4 ⊢ (¬ 𝑥 = 𝑦 → ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅)))
74	2, 73	pm2.61i 175	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
75	74	ralrimivva 2954	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
76		fveq2 6103	. . 3 ⊢ (𝑥 = 𝑦 → (𝑀‘𝑥) = (𝑀‘𝑦))
77	76	disjor 4567	. 2 ⊢ (Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥 = 𝑦 ∨ ((𝑀‘𝑥) ∩ (𝑀‘𝑦)) = ∅))
78	75, 77	sylibr 223	1 ⊢ (𝑉 ∈ 𝑊 → Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539  ∅c0 3874  ⟨cotp 4133  Disj wdisj 4553   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   2SPathsOt c2spthot 26383

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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847

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-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  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-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059  df-2nd 7060

This theorem is referenced by:  usgreghash2spot  26596