Theorem relop 4599

Description: A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
relop.1	⊢ 𝐴 ∈ V
relop.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
relop	⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem relop

Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rel 4459	. 2 ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩ ⊆ (V × V))
2		dfss2 3015	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
3		vex 2623	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
4		relop.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
5		relop.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
6	3, 4, 5	elop 4067	. . . . . . . . 9 ⊢ (𝑧 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}))
7		elvv 4513	. . . . . . . . 9 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)
8	6, 7	imbi12i 238	. . . . . . . 8 ⊢ ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩))
9		jaob 667	. . . . . . . 8 ⊢ (((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
10	8, 9	bitri 183	. . . . . . 7 ⊢ ((𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
11	10	albii 1405	. . . . . 6 ⊢ (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ ∀𝑧((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
12		19.26 1416	. . . . . 6 ⊢ (∀𝑧((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ (𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
13	11, 12	bitri 183	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
14	2, 13	bitri 183	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)))
15	4	snex 4026	. . . . . . 7 ⊢ {𝐴} ∈ V
16		eqeq1 2095	. . . . . . . 8 ⊢ (𝑧 = {𝐴} → (𝑧 = {𝐴} ↔ {𝐴} = {𝐴}))
17		eqeq1 2095	. . . . . . . . . 10 ⊢ (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴} = ⟨𝑥, 𝑦⟩))
18		eqcom 2091	. . . . . . . . . . 11 ⊢ ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴})
19		vex 2623	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
20		vex 2623	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
21	19, 20, 4	opeqsn 4088	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ = {𝐴} ↔ (𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))
22	18, 21	bitri 183	. . . . . . . . . 10 ⊢ ({𝐴} = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))
23	17, 22	syl6bb 195	. . . . . . . . 9 ⊢ (𝑧 = {𝐴} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
24	23	2exbidv 1797	. . . . . . . 8 ⊢ (𝑧 = {𝐴} → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
25	16, 24	imbi12d 233	. . . . . . 7 ⊢ (𝑧 = {𝐴} → ((𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))))
26	15, 25	spcv 2713	. . . . . 6 ⊢ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
27		sneq 3461	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
28	27	eqeq2d 2100	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝐴 = {𝑤} ↔ 𝐴 = {𝑥}))
29	28	cbvexv 1844	. . . . . . 7 ⊢ (∃𝑤 𝐴 = {𝑤} ↔ ∃𝑥 𝐴 = {𝑥})
30		a9ev 1633	. . . . . . . . . 10 ⊢ ∃𝑦 𝑦 = 𝑥
31		equcom 1640	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
32	31	exbii 1542	. . . . . . . . . 10 ⊢ (∃𝑦 𝑦 = 𝑥 ↔ ∃𝑦 𝑥 = 𝑦)
33	30, 32	mpbi 144	. . . . . . . . 9 ⊢ ∃𝑦 𝑥 = 𝑦
34		19.41v 1831	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ (∃𝑦 𝑥 = 𝑦 ∧ 𝐴 = {𝑥}))
35	33, 34	mpbiran 887	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ 𝐴 = {𝑥})
36	35	exbii 1542	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ ∃𝑥 𝐴 = {𝑥})
37		eqid 2089	. . . . . . . 8 ⊢ {𝐴} = {𝐴}
38	37	a1bi 242	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥}) ↔ ({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})))
39	29, 36, 38	3bitr2ri 208	. . . . . 6 ⊢ (({𝐴} = {𝐴} → ∃𝑥∃𝑦(𝑥 = 𝑦 ∧ 𝐴 = {𝑥})) ↔ ∃𝑤 𝐴 = {𝑤})
40	26, 39	sylib 121	. . . . 5 ⊢ (∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑤 𝐴 = {𝑤})
41		eqid 2089	. . . . . 6 ⊢ {𝐴, 𝐵} = {𝐴, 𝐵}
42		prexg 4047	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
43	4, 5, 42	mp2an 418	. . . . . . 7 ⊢ {𝐴, 𝐵} ∈ V
44		eqeq1 2095	. . . . . . . 8 ⊢ (𝑧 = {𝐴, 𝐵} → (𝑧 = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴, 𝐵}))
45		eqeq1 2095	. . . . . . . . 9 ⊢ (𝑧 = {𝐴, 𝐵} → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ {𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
46	45	2exbidv 1797	. . . . . . . 8 ⊢ (𝑧 = {𝐴, 𝐵} → (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
47	44, 46	imbi12d 233	. . . . . . 7 ⊢ (𝑧 = {𝐴, 𝐵} → ((𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ↔ ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)))
48	43, 47	spcv 2713	. . . . . 6 ⊢ (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ({𝐴, 𝐵} = {𝐴, 𝐵} → ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩))
49	41, 48	mpi 15	. . . . 5 ⊢ (∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) → ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩)
50		eqcom 2091	. . . . . . . . . 10 ⊢ ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = {𝐴, 𝐵})
51	19, 20, 4, 5	opeqpr 4089	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦⟩ = {𝐴, 𝐵} ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
52	50, 51	bitri 183	. . . . . . . . 9 ⊢ ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ ↔ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})))
53		idd 21	. . . . . . . . . 10 ⊢ (𝐴 = {𝑤} → ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
54		eqtr2 2107	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → {𝑥, 𝑦} = {𝑤})
55		vex 2623	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
56	19, 20, 55	preqsn 3625	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} = {𝑤} ↔ (𝑥 = 𝑦 ∧ 𝑦 = 𝑤))
57	56	simplbi 269	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} = {𝑤} → 𝑥 = 𝑦)
58	54, 57	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → 𝑥 = 𝑦)
59		dfsn2 3464	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑥} = {𝑥, 𝑥}
60		preq2 3524	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑥} = {𝑥, 𝑦})
61	59, 60	syl5req 2134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → {𝑥, 𝑦} = {𝑥})
62	61	eqeq2d 2100	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} ↔ 𝐴 = {𝑥}))
63	59, 60	syl5eq 2133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑥, 𝑦})
64	63	eqeq2d 2100	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝐵 = {𝑥} ↔ 𝐵 = {𝑥, 𝑦}))
65	62, 64	anbi12d 458	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) ↔ (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
66	65	biimpd 143	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
67	66	expd 255	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
68	67	com12 30	. . . . . . . . . . . . . 14 ⊢ (𝐴 = {𝑥, 𝑦} → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
69	68	adantr 271	. . . . . . . . . . . . 13 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝑥 = 𝑦 → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
70	58, 69	mpd 13	. . . . . . . . . . . 12 ⊢ ((𝐴 = {𝑥, 𝑦} ∧ 𝐴 = {𝑤}) → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
71	70	expcom 115	. . . . . . . . . . 11 ⊢ (𝐴 = {𝑤} → (𝐴 = {𝑥, 𝑦} → (𝐵 = {𝑥} → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))))
72	71	impd 252	. . . . . . . . . 10 ⊢ (𝐴 = {𝑤} → ((𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥}) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
73	53, 72	jaod 673	. . . . . . . . 9 ⊢ (𝐴 = {𝑤} → (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∨ (𝐴 = {𝑥, 𝑦} ∧ 𝐵 = {𝑥})) → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
74	52, 73	syl5bi 151	. . . . . . . 8 ⊢ (𝐴 = {𝑤} → ({𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → (𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
75	74	2eximdv 1811	. . . . . . 7 ⊢ (𝐴 = {𝑤} → (∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
76	75	exlimiv 1535	. . . . . 6 ⊢ (∃𝑤 𝐴 = {𝑤} → (∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})))
77	76	imp 123	. . . . 5 ⊢ ((∃𝑤 𝐴 = {𝑤} ∧ ∃𝑥∃𝑦{𝐴, 𝐵} = ⟨𝑥, 𝑦⟩) → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
78	40, 49, 77	syl2an 284	. . . 4 ⊢ ((∀𝑧(𝑧 = {𝐴} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩) ∧ ∀𝑧(𝑧 = {𝐴, 𝐵} → ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩)) → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
79	14, 78	sylbi 120	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) → ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
80		simpr 109	. . . . . . . . . . 11 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = {𝐴})
81		equid 1635	. . . . . . . . . . . . . 14 ⊢ 𝑥 = 𝑥
82	81	jctl 308	. . . . . . . . . . . . 13 ⊢ (𝐴 = {𝑥} → (𝑥 = 𝑥 ∧ 𝐴 = {𝑥}))
83	19, 19, 4	opeqsn 4088	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑥⟩ = {𝐴} ↔ (𝑥 = 𝑥 ∧ 𝐴 = {𝑥}))
84	82, 83	sylibr 133	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥} → ⟨𝑥, 𝑥⟩ = {𝐴})
85	84	adantr 271	. . . . . . . . . . 11 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ⟨𝑥, 𝑥⟩ = {𝐴})
86	80, 85	eqtr4d 2124	. . . . . . . . . 10 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → 𝑧 = ⟨𝑥, 𝑥⟩)
87		opeq12 3630	. . . . . . . . . . . 12 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑥) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑥⟩)
88	87	eqeq2d 2100	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑥) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑥⟩))
89	19, 19, 88	spc2ev 2715	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑥⟩ → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
90	86, 89	syl 14	. . . . . . . . 9 ⊢ ((𝐴 = {𝑥} ∧ 𝑧 = {𝐴}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
91	90	adantlr 462	. . . . . . . 8 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
92		preq12 3525	. . . . . . . . . . . 12 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → {𝐴, 𝐵} = {{𝑥}, {𝑥, 𝑦}})
93	92	eqeq2d 2100	. . . . . . . . . . 11 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 = {𝐴, 𝐵} ↔ 𝑧 = {{𝑥}, {𝑥, 𝑦}}))
94	93	biimpa 291	. . . . . . . . . 10 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = {{𝑥}, {𝑥, 𝑦}})
95	19, 20	dfop 3627	. . . . . . . . . 10 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
96	94, 95	syl6eqr 2139	. . . . . . . . 9 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → 𝑧 = ⟨𝑥, 𝑦⟩)
97		opeq12 3630	. . . . . . . . . . 11 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨𝑤, 𝑣⟩ = ⟨𝑥, 𝑦⟩)
98	97	eqeq2d 2100	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
99	19, 20, 98	spc2ev 2715	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
100	96, 99	syl 14	. . . . . . . 8 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ 𝑧 = {𝐴, 𝐵}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
101	91, 100	jaodan 747	. . . . . . 7 ⊢ (((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) ∧ (𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵})) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
102	101	ex 114	. . . . . 6 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ((𝑧 = {𝐴} ∨ 𝑧 = {𝐴, 𝐵}) → ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩))
103		elvv 4513	. . . . . 6 ⊢ (𝑧 ∈ (V × V) ↔ ∃𝑤∃𝑣 𝑧 = ⟨𝑤, 𝑣⟩)
104	102, 6, 103	3imtr4g 204	. . . . 5 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → (𝑧 ∈ ⟨𝐴, 𝐵⟩ → 𝑧 ∈ (V × V)))
105	104	ssrdv 3032	. . . 4 ⊢ ((𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
106	105	exlimivv 1825	. . 3 ⊢ (∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}) → ⟨𝐴, 𝐵⟩ ⊆ (V × V))
107	79, 106	impbii 125	. 2 ⊢ (⟨𝐴, 𝐵⟩ ⊆ (V × V) ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
108	1, 107	bitri 183	1 ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 665  ∀wal 1288   = wceq 1290  ∃wex 1427   ∈ wcel 1439  Vcvv 2620   ⊆ wss 3000  {csn 3450  {cpr 3451  ⟨cop 3453   × cxp 4450  Rel wrel 4457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-xp 4458  df-rel 4459

This theorem is referenced by:  funopg  5061