Theorem oqelins4 5795

Description: Ordered quadruple membership in Ins4. (Contributed by SF, 13-Feb-2015.)

Hypothesis

Ref	Expression
oqelins4.4	⊢ D ∈ V

Assertion

Ref	Expression
oqelins4	⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R)

Proof of Theorem oqelins4

Dummy variables a b p x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. . 3 ⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R → ⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ V)
2		opexb 4604	. . . . 5 ⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ V ↔ (A ∈ V ∧ ⟨B, ⟨C, D⟩⟩ ∈ V))
3		opexb 4604	. . . . . 6 ⊢ (⟨B, ⟨C, D⟩⟩ ∈ V ↔ (B ∈ V ∧ ⟨C, D⟩ ∈ V))
4	3	anbi2i 675	. . . . 5 ⊢ ((A ∈ V ∧ ⟨B, ⟨C, D⟩⟩ ∈ V) ↔ (A ∈ V ∧ (B ∈ V ∧ ⟨C, D⟩ ∈ V)))
5	2, 4	bitri 240	. . . 4 ⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ V ↔ (A ∈ V ∧ (B ∈ V ∧ ⟨C, D⟩ ∈ V)))
6		opexb 4604	. . . . . . 7 ⊢ (⟨C, D⟩ ∈ V ↔ (C ∈ V ∧ D ∈ V))
7	6	simplbi 446	. . . . . 6 ⊢ (⟨C, D⟩ ∈ V → C ∈ V)
8	7	anim2i 552	. . . . 5 ⊢ ((B ∈ V ∧ ⟨C, D⟩ ∈ V) → (B ∈ V ∧ C ∈ V))
9	8	anim2i 552	. . . 4 ⊢ ((A ∈ V ∧ (B ∈ V ∧ ⟨C, D⟩ ∈ V)) → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
10	5, 9	sylbi 187	. . 3 ⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ V → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
11	1, 10	syl 15	. 2 ⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
12		elex 2868	. . 3 ⊢ (⟨A, ⟨B, C⟩⟩ ∈ R → ⟨A, ⟨B, C⟩⟩ ∈ V)
13		opexb 4604	. . . 4 ⊢ (⟨A, ⟨B, C⟩⟩ ∈ V ↔ (A ∈ V ∧ ⟨B, C⟩ ∈ V))
14		opexb 4604	. . . . 5 ⊢ (⟨B, C⟩ ∈ V ↔ (B ∈ V ∧ C ∈ V))
15	14	anbi2i 675	. . . 4 ⊢ ((A ∈ V ∧ ⟨B, C⟩ ∈ V) ↔ (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
16	13, 15	bitri 240	. . 3 ⊢ (⟨A, ⟨B, C⟩⟩ ∈ V ↔ (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
17	12, 16	sylib 188	. 2 ⊢ (⟨A, ⟨B, C⟩⟩ ∈ R → (A ∈ V ∧ (B ∈ V ∧ C ∈ V)))
18		opeq1 4579	. . . . . . 7 ⊢ (x = A → ⟨x, ⟨B, ⟨C, D⟩⟩⟩ = ⟨A, ⟨B, ⟨C, D⟩⟩⟩)
19	18	eleq1d 2419	. . . . . 6 ⊢ (x = A → (⟨x, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R))
20		opeq1 4579	. . . . . . 7 ⊢ (x = A → ⟨x, ⟨B, C⟩⟩ = ⟨A, ⟨B, C⟩⟩)
21	20	eleq1d 2419	. . . . . 6 ⊢ (x = A → (⟨x, ⟨B, C⟩⟩ ∈ R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R))
22	19, 21	bibi12d 312	. . . . 5 ⊢ (x = A → ((⟨x, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, C⟩⟩ ∈ R) ↔ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R)))
23	22	imbi2d 307	. . . 4 ⊢ (x = A → (((B ∈ V ∧ C ∈ V) → (⟨x, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, C⟩⟩ ∈ R)) ↔ ((B ∈ V ∧ C ∈ V) → (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R))))
24		opeq1 4579	. . . . . . . 8 ⊢ (y = B → ⟨y, ⟨z, D⟩⟩ = ⟨B, ⟨z, D⟩⟩)
25	24	opeq2d 4586	. . . . . . 7 ⊢ (y = B → ⟨x, ⟨y, ⟨z, D⟩⟩⟩ = ⟨x, ⟨B, ⟨z, D⟩⟩⟩)
26	25	eleq1d 2419	. . . . . 6 ⊢ (y = B → (⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, ⟨z, D⟩⟩⟩ ∈ Ins4 R))
27		opeq1 4579	. . . . . . . 8 ⊢ (y = B → ⟨y, z⟩ = ⟨B, z⟩)
28	27	opeq2d 4586	. . . . . . 7 ⊢ (y = B → ⟨x, ⟨y, z⟩⟩ = ⟨x, ⟨B, z⟩⟩)
29	28	eleq1d 2419	. . . . . 6 ⊢ (y = B → (⟨x, ⟨y, z⟩⟩ ∈ R ↔ ⟨x, ⟨B, z⟩⟩ ∈ R))
30	26, 29	bibi12d 312	. . . . 5 ⊢ (y = B → ((⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨y, z⟩⟩ ∈ R) ↔ (⟨x, ⟨B, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, z⟩⟩ ∈ R)))
31		opeq1 4579	. . . . . . . . 9 ⊢ (z = C → ⟨z, D⟩ = ⟨C, D⟩)
32	31	opeq2d 4586	. . . . . . . 8 ⊢ (z = C → ⟨B, ⟨z, D⟩⟩ = ⟨B, ⟨C, D⟩⟩)
33	32	opeq2d 4586	. . . . . . 7 ⊢ (z = C → ⟨x, ⟨B, ⟨z, D⟩⟩⟩ = ⟨x, ⟨B, ⟨C, D⟩⟩⟩)
34	33	eleq1d 2419	. . . . . 6 ⊢ (z = C → (⟨x, ⟨B, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R))
35		opeq2 4580	. . . . . . . 8 ⊢ (z = C → ⟨B, z⟩ = ⟨B, C⟩)
36	35	opeq2d 4586	. . . . . . 7 ⊢ (z = C → ⟨x, ⟨B, z⟩⟩ = ⟨x, ⟨B, C⟩⟩)
37	36	eleq1d 2419	. . . . . 6 ⊢ (z = C → (⟨x, ⟨B, z⟩⟩ ∈ R ↔ ⟨x, ⟨B, C⟩⟩ ∈ R))
38	34, 37	bibi12d 312	. . . . 5 ⊢ (z = C → ((⟨x, ⟨B, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, z⟩⟩ ∈ R) ↔ (⟨x, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, C⟩⟩ ∈ R)))
39		df-ins4 5757	. . . . . . 7 ⊢ Ins4 R = (◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))) “ R)
40	39	eleq2i 2417	. . . . . 6 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ (◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))) “ R))
41		brcnv 4893	. . . . . . . . 9 ⊢ (p◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))⟨x, ⟨y, ⟨z, D⟩⟩⟩ ↔ ⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))p)
42		brtxp 5784	. . . . . . . . . 10 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))p ↔ ∃a∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b))
43		3ancoma 941	. . . . . . . . . . . . . 14 ⊢ ((p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b))
44		3anass 938	. . . . . . . . . . . . . 14 ⊢ ((⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ (p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
45		vex 2863	. . . . . . . . . . . . . . . . 17 ⊢ x ∈ V
46		vex 2863	. . . . . . . . . . . . . . . . . 18 ⊢ y ∈ V
47		vex 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ z ∈ V
48		oqelins4.4	. . . . . . . . . . . . . . . . . . 19 ⊢ D ∈ V
49	47, 48	opex 4589	. . . . . . . . . . . . . . . . . 18 ⊢ ⟨z, D⟩ ∈ V
50	46, 49	opex 4589	. . . . . . . . . . . . . . . . 17 ⊢ ⟨y, ⟨z, D⟩⟩ ∈ V
51	45, 50	opbr1st 5502	. . . . . . . . . . . . . . . 16 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ↔ x = a)
52		equcom 1680	. . . . . . . . . . . . . . . 16 ⊢ (x = a ↔ a = x)
53	51, 52	bitri 240	. . . . . . . . . . . . . . 15 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ↔ a = x)
54	53	anbi1i 676	. . . . . . . . . . . . . 14 ⊢ ((⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ (p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)) ↔ (a = x ∧ (p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
55	43, 44, 54	3bitri 262	. . . . . . . . . . . . 13 ⊢ ((p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (a = x ∧ (p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
56	55	exbii 1582	. . . . . . . . . . . 12 ⊢ (∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ ∃b(a = x ∧ (p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
57		19.42v 1905	. . . . . . . . . . . 12 ⊢ (∃b(a = x ∧ (p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)) ↔ (a = x ∧ ∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
58	56, 57	bitri 240	. . . . . . . . . . 11 ⊢ (∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (a = x ∧ ∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
59	58	exbii 1582	. . . . . . . . . 10 ⊢ (∃a∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩1st a ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ ∃a(a = x ∧ ∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
60		opeq1 4579	. . . . . . . . . . . . . 14 ⊢ (a = x → ⟨a, b⟩ = ⟨x, b⟩)
61	60	eqeq2d 2364	. . . . . . . . . . . . 13 ⊢ (a = x → (p = ⟨a, b⟩ ↔ p = ⟨x, b⟩))
62	61	anbi1d 685	. . . . . . . . . . . 12 ⊢ (a = x → ((p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
63	62	exbidv 1626	. . . . . . . . . . 11 ⊢ (a = x → (∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ ∃b(p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)))
64	45, 63	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃a(a = x ∧ ∃b(p = ⟨a, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b)) ↔ ∃b(p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b))
65	42, 59, 64	3bitri 262	. . . . . . . . 9 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))p ↔ ∃b(p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b))
66		ancom 437	. . . . . . . . . . . 12 ⊢ ((p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b ∧ p = ⟨x, b⟩))
67		brtxp 5784	. . . . . . . . . . . . . 14 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b ↔ ∃p∃a(b = ⟨p, a⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a))
68		3anrot 939	. . . . . . . . . . . . . . . 16 ⊢ ((b = ⟨p, a⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a) ↔ (⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a ∧ b = ⟨p, a⟩))
69	45, 50	brco2nd 5779	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ↔ ⟨y, ⟨z, D⟩⟩1st p)
70	46, 49	opbr1st 5502	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨y, ⟨z, D⟩⟩1st p ↔ y = p)
71		equcom 1680	. . . . . . . . . . . . . . . . . 18 ⊢ (y = p ↔ p = y)
72	69, 70, 71	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ↔ p = y)
73	45, 50	brco2nd 5779	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a ↔ ⟨y, ⟨z, D⟩⟩(1st ∘ 2nd )a)
74	46, 49	brco2nd 5779	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨y, ⟨z, D⟩⟩(1st ∘ 2nd )a ↔ ⟨z, D⟩1st a)
75	47, 48	opbr1st 5502	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨z, D⟩1st a ↔ z = a)
76	74, 75	bitri 240	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨y, ⟨z, D⟩⟩(1st ∘ 2nd )a ↔ z = a)
77		equcom 1680	. . . . . . . . . . . . . . . . . 18 ⊢ (z = a ↔ a = z)
78	73, 76, 77	3bitri 262	. . . . . . . . . . . . . . . . 17 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a ↔ a = z)
79		biid 227	. . . . . . . . . . . . . . . . 17 ⊢ (b = ⟨p, a⟩ ↔ b = ⟨p, a⟩)
80	72, 78, 79	3anbi123i 1140	. . . . . . . . . . . . . . . 16 ⊢ ((⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a ∧ b = ⟨p, a⟩) ↔ (p = y ∧ a = z ∧ b = ⟨p, a⟩))
81	68, 80	bitri 240	. . . . . . . . . . . . . . 15 ⊢ ((b = ⟨p, a⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a) ↔ (p = y ∧ a = z ∧ b = ⟨p, a⟩))
82	81	2exbii 1583	. . . . . . . . . . . . . 14 ⊢ (∃p∃a(b = ⟨p, a⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩(1st ∘ 2nd )p ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ∘ 2nd )a) ↔ ∃p∃a(p = y ∧ a = z ∧ b = ⟨p, a⟩))
83		opeq1 4579	. . . . . . . . . . . . . . . 16 ⊢ (p = y → ⟨p, a⟩ = ⟨y, a⟩)
84	83	eqeq2d 2364	. . . . . . . . . . . . . . 15 ⊢ (p = y → (b = ⟨p, a⟩ ↔ b = ⟨y, a⟩))
85		opeq2 4580	. . . . . . . . . . . . . . . 16 ⊢ (a = z → ⟨y, a⟩ = ⟨y, z⟩)
86	85	eqeq2d 2364	. . . . . . . . . . . . . . 15 ⊢ (a = z → (b = ⟨y, a⟩ ↔ b = ⟨y, z⟩))
87	46, 47, 84, 86	ceqsex2v 2897	. . . . . . . . . . . . . 14 ⊢ (∃p∃a(p = y ∧ a = z ∧ b = ⟨p, a⟩) ↔ b = ⟨y, z⟩)
88	67, 82, 87	3bitri 262	. . . . . . . . . . . . 13 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b ↔ b = ⟨y, z⟩)
89	88	anbi1i 676	. . . . . . . . . . . 12 ⊢ ((⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b ∧ p = ⟨x, b⟩) ↔ (b = ⟨y, z⟩ ∧ p = ⟨x, b⟩))
90	66, 89	bitri 240	. . . . . . . . . . 11 ⊢ ((p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ (b = ⟨y, z⟩ ∧ p = ⟨x, b⟩))
91	90	exbii 1582	. . . . . . . . . 10 ⊢ (∃b(p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ ∃b(b = ⟨y, z⟩ ∧ p = ⟨x, b⟩))
92	46, 47	opex 4589	. . . . . . . . . . 11 ⊢ ⟨y, z⟩ ∈ V
93		opeq2 4580	. . . . . . . . . . . 12 ⊢ (b = ⟨y, z⟩ → ⟨x, b⟩ = ⟨x, ⟨y, z⟩⟩)
94	93	eqeq2d 2364	. . . . . . . . . . 11 ⊢ (b = ⟨y, z⟩ → (p = ⟨x, b⟩ ↔ p = ⟨x, ⟨y, z⟩⟩))
95	92, 94	ceqsexv 2895	. . . . . . . . . 10 ⊢ (∃b(b = ⟨y, z⟩ ∧ p = ⟨x, b⟩) ↔ p = ⟨x, ⟨y, z⟩⟩)
96	91, 95	bitri 240	. . . . . . . . 9 ⊢ (∃b(p = ⟨x, b⟩ ∧ ⟨x, ⟨y, ⟨z, D⟩⟩⟩((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))b) ↔ p = ⟨x, ⟨y, z⟩⟩)
97	41, 65, 96	3bitri 262	. . . . . . . 8 ⊢ (p◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))⟨x, ⟨y, ⟨z, D⟩⟩⟩ ↔ p = ⟨x, ⟨y, z⟩⟩)
98	97	rexbii 2640	. . . . . . 7 ⊢ (∃p ∈ R p◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))⟨x, ⟨y, ⟨z, D⟩⟩⟩ ↔ ∃p ∈ R p = ⟨x, ⟨y, z⟩⟩)
99		elima 4755	. . . . . . 7 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ (◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))) “ R) ↔ ∃p ∈ R p◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd )))⟨x, ⟨y, ⟨z, D⟩⟩⟩)
100		risset 2662	. . . . . . 7 ⊢ (⟨x, ⟨y, z⟩⟩ ∈ R ↔ ∃p ∈ R p = ⟨x, ⟨y, z⟩⟩)
101	98, 99, 100	3bitr4i 268	. . . . . 6 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ (◡(1st ⊗ ((1st ∘ 2nd ) ⊗ ((1st ∘ 2nd ) ∘ 2nd ))) “ R) ↔ ⟨x, ⟨y, z⟩⟩ ∈ R)
102	40, 101	bitri 240	. . . . 5 ⊢ (⟨x, ⟨y, ⟨z, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨y, z⟩⟩ ∈ R)
103	30, 38, 102	vtocl2g 2919	. . . 4 ⊢ ((B ∈ V ∧ C ∈ V) → (⟨x, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨x, ⟨B, C⟩⟩ ∈ R))
104	23, 103	vtoclg 2915	. . 3 ⊢ (A ∈ V → ((B ∈ V ∧ C ∈ V) → (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R)))
105	104	imp 418	. 2 ⊢ ((A ∈ V ∧ (B ∈ V ∧ C ∈ V)) → (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R))
106	11, 17, 105	pm5.21nii 342	1 ⊢ (⟨A, ⟨B, ⟨C, D⟩⟩⟩ ∈ Ins4 R ↔ ⟨A, ⟨B, C⟩⟩ ∈ R)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   ∘ ccom 4722   “ cima 4723  ^◡ccnv 4772  2^nd c2nd 4784   ⊗ ctxp 5736   Ins4 cins4 5756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-cnv 4786  df-2nd 4798  df-txp 5737  df-ins4 5757

This theorem is referenced by:  composeex  5821  addcfnex  5825  funsex  5829  crossex  5851  domfnex  5871  ranfnex  5872  transex  5911  antisymex  5913  connexex  5914  foundex  5915  extex  5916  symex  5917  ovmuc  6131  mucex  6134  ovcelem1  6172  ceex  6175