4fvwrd4 - Intuitionistic Logic Explorer

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Theorem 4fvwrd4 10525

Description: The first four function values of a word of length at least 4. (Contributed by Alexander van der Vekens, 18-Nov-2017.)

**Assertion**
Ref	Expression
4fvwrd4	⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))

Distinct variable groups: 𝑃,𝑎,𝑏,𝑐,𝑑 𝑉,𝑎,𝑏,𝑐,𝑑 Allowed substitution hints: 𝐿(𝑎,𝑏,𝑐,𝑑)

**Proof of Theorem 4fvwrd4**
Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 𝑃:(0...𝐿)⟶𝑉)
2		0nn0 9557	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
3		elnn0uz 9939	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ ↔ 0 ∈ (ℤ_≥‘0))
4	2, 3	mpbi 145	. . . . . . . 8 ⊢ 0 ∈ (ℤ_≥‘0)
5		3nn0 9560	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
6		elnn0uz 9939	. . . . . . . . . . 11 ⊢ (3 ∈ ℕ₀ ↔ 3 ∈ (ℤ_≥‘0))
7	5, 6	mpbi 145	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘0)
8		uzss 9922	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘0) → (ℤ_≥‘3) ⊆ (ℤ_≥‘0))
9	7, 8	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘3) ⊆ (ℤ_≥‘0)
10	9	sseli 3244	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘0))
11		eluzfz 10402	. . . . . . . 8 ⊢ ((0 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘0)) → 0 ∈ (0...𝐿))
12	4, 10, 11	sylancr 418	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 0 ∈ (0...𝐿))
13	12	adantr 276	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 0 ∈ (0...𝐿))
14	1, 13	ffvelcdmd 5835	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘0) ∈ 𝑉)
15		risset 2578	. . . . . 6 ⊢ ((𝑃‘0) ∈ 𝑉 ↔ ∃𝑎 ∈ 𝑉 𝑎 = (𝑃‘0))
16		eqcom 2240	. . . . . . 7 ⊢ (𝑎 = (𝑃‘0) ↔ (𝑃‘0) = 𝑎)
17	16	rexbii 2557	. . . . . 6 ⊢ (∃𝑎 ∈ 𝑉 𝑎 = (𝑃‘0) ↔ ∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎)
18	15, 17	bitri 184	. . . . 5 ⊢ ((𝑃‘0) ∈ 𝑉 ↔ ∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎)
19	14, 18	sylib 122	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎)
20		1eluzge0 9953	. . . . . . . 8 ⊢ 1 ∈ (ℤ_≥‘0)
21		1z 9649	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
22		3z 9652	. . . . . . . . . . 11 ⊢ 3 ∈ ℤ
23		1le3 9495	. . . . . . . . . . 11 ⊢ 1 ≤ 3
24		eluz2 9906	. . . . . . . . . . 11 ⊢ (3 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 3 ∈ ℤ ∧ 1 ≤ 3))
25	21, 22, 23, 24	mpbir3an 1210	. . . . . . . . . 10 ⊢ 3 ∈ (ℤ_≥‘1)
26		uzss 9922	. . . . . . . . . 10 ⊢ (3 ∈ (ℤ_≥‘1) → (ℤ_≥‘3) ⊆ (ℤ_≥‘1))
27	25, 26	ax-mp 5	. . . . . . . . 9 ⊢ (ℤ_≥‘3) ⊆ (ℤ_≥‘1)
28	27	sseli 3244	. . . . . . . 8 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘1))
29		eluzfz 10402	. . . . . . . 8 ⊢ ((1 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘1)) → 1 ∈ (0...𝐿))
30	20, 28, 29	sylancr 418	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 1 ∈ (0...𝐿))
31	30	adantr 276	. . . . . 6 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 1 ∈ (0...𝐿))
32	1, 31	ffvelcdmd 5835	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘1) ∈ 𝑉)
33		risset 2578	. . . . . 6 ⊢ ((𝑃‘1) ∈ 𝑉 ↔ ∃𝑏 ∈ 𝑉 𝑏 = (𝑃‘1))
34		eqcom 2240	. . . . . . 7 ⊢ (𝑏 = (𝑃‘1) ↔ (𝑃‘1) = 𝑏)
35	34	rexbii 2557	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑉 𝑏 = (𝑃‘1) ↔ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏)
36	33, 35	bitri 184	. . . . 5 ⊢ ((𝑃‘1) ∈ 𝑉 ↔ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏)
37	32, 36	sylib 122	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏)
38	19, 37	jca 306	. . 3 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏))
39		2eluzge0 9954	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
40		uzuzle23 9941	. . . . . . 7 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 𝐿 ∈ (ℤ_≥‘2))
41		eluzfz 10402	. . . . . . 7 ⊢ ((2 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘2)) → 2 ∈ (0...𝐿))
42	39, 40, 41	sylancr 418	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 2 ∈ (0...𝐿))
43	42	adantr 276	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 2 ∈ (0...𝐿))
44	1, 43	ffvelcdmd 5835	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘2) ∈ 𝑉)
45		risset 2578	. . . . 5 ⊢ ((𝑃‘2) ∈ 𝑉 ↔ ∃𝑐 ∈ 𝑉 𝑐 = (𝑃‘2))
46		eqcom 2240	. . . . . 6 ⊢ (𝑐 = (𝑃‘2) ↔ (𝑃‘2) = 𝑐)
47	46	rexbii 2557	. . . . 5 ⊢ (∃𝑐 ∈ 𝑉 𝑐 = (𝑃‘2) ↔ ∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐)
48	45, 47	bitri 184	. . . 4 ⊢ ((𝑃‘2) ∈ 𝑉 ↔ ∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐)
49	44, 48	sylib 122	. . 3 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐)
50		eluzfz 10402	. . . . . . 7 ⊢ ((3 ∈ (ℤ_≥‘0) ∧ 𝐿 ∈ (ℤ_≥‘3)) → 3 ∈ (0...𝐿))
51	7, 50	mpan 428	. . . . . 6 ⊢ (𝐿 ∈ (ℤ_≥‘3) → 3 ∈ (0...𝐿))
52	51	adantr 276	. . . . 5 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → 3 ∈ (0...𝐿))
53	1, 52	ffvelcdmd 5835	. . . 4 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → (𝑃‘3) ∈ 𝑉)
54		risset 2578	. . . . 5 ⊢ ((𝑃‘3) ∈ 𝑉 ↔ ∃𝑑 ∈ 𝑉 𝑑 = (𝑃‘3))
55		eqcom 2240	. . . . . 6 ⊢ (𝑑 = (𝑃‘3) ↔ (𝑃‘3) = 𝑑)
56	55	rexbii 2557	. . . . 5 ⊢ (∃𝑑 ∈ 𝑉 𝑑 = (𝑃‘3) ↔ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)
57	54, 56	bitri 184	. . . 4 ⊢ ((𝑃‘3) ∈ 𝑉 ↔ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)
58	53, 57	sylib 122	. . 3 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)
59	38, 49, 58	jca32 310	. 2 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
60		r19.42v 2708	. . . . . 6 ⊢ (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑑 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
61		r19.42v 2708	. . . . . . 7 ⊢ (∃𝑑 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) ↔ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑))
62	61	anbi2i 461	. . . . . 6 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑑 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
63	60, 62	bitri 184	. . . . 5 ⊢ (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
64	63	rexbii 2557	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ ∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
65	64	2rexbii 2559	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
66		r19.42v 2708	. . . . 5 ⊢ (∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑐 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
67		r19.41v 2707	. . . . . 6 ⊢ (∃𝑐 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑) ↔ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑))
68	67	anbi2i 461	. . . . 5 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ∃𝑐 ∈ 𝑉 ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
69	66, 68	bitri 184	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
70	69	2rexbii 2559	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
71		r19.41v 2707	. . . . . 6 ⊢ (∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (∃𝑏 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
72		r19.42v 2708	. . . . . . 7 ⊢ (∃𝑏 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ↔ ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏))
73	72	anbi1i 462	. . . . . 6 ⊢ ((∃𝑏 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
74	71, 73	bitri 184	. . . . 5 ⊢ (∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
75	74	rexbii 2557	. . . 4 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ∃𝑎 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
76		r19.41v 2707	. . . 4 ⊢ (∃𝑎 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ (∃𝑎 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
77		r19.41v 2707	. . . . 5 ⊢ (∃𝑎 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ↔ (∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏))
78	77	anbi1i 462	. . . 4 ⊢ ((∃𝑎 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
79	75, 76, 78	3bitri 206	. . 3 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)) ↔ ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
80	65, 70, 79	3bitri 206	. 2 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) ↔ ((∃𝑎 ∈ 𝑉 (𝑃‘0) = 𝑎 ∧ ∃𝑏 ∈ 𝑉 (𝑃‘1) = 𝑏) ∧ (∃𝑐 ∈ 𝑉 (𝑃‘2) = 𝑐 ∧ ∃𝑑 ∈ 𝑉 (𝑃‘3) = 𝑑)))
81	59, 80	sylibr 134	1 ⊢ ((𝐿 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...𝐿)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 ⊆ wss 3220 class class class wbr 4125 ⟶wf 5368 ‘cfv 5372 (class class class)co 6075 0cc0 8169 1c1 8170 ≤ cle 8351 2c2 9334 3c3 9335 ℕ₀cn0 9542 ℤcz 9623 ℤ_≥cuz 9900 ...cfz 10390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-addcom 8269 ax-addass 8271 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-0id 8277 ax-rnegex 8278 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-ltadd 8285

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-inn 9284 df-2 9342 df-3 9343 df-n0 9543 df-z 9624 df-uz 9901 df-fz 10391

This theorem is referenced by: (None)

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