Theorem djuunxp 10039

Description: The union of a disjoint union and its inversion is the Cartesian product of an unordered pair and the union of the left and right classes of the disjoint unions. (Proposed by GL, 4-Jul-2022.) (Contributed by AV, 4-Jul-2022.)

Assertion

Ref	Expression
djuunxp	⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵))

Proof of Theorem djuunxp

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

Step	Hyp	Ref	Expression
1		djuss 10038	. . 3 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		djuss 10038	. . . 4 ⊢ (𝐵 ⊔ 𝐴) ⊆ ({∅, 1o} × (𝐵 ∪ 𝐴))
3		uncom 4160	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
4	3	xpeq2i 5685	. . . 4 ⊢ ({∅, 1o} × (𝐴 ∪ 𝐵)) = ({∅, 1o} × (𝐵 ∪ 𝐴))
5	2, 4	sseqtrri 3962	. . 3 ⊢ (𝐵 ⊔ 𝐴) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
6	1, 5	unssi 4192	. 2 ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
7		elxpi 5680	. . . 4 ⊢ (𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))))
8		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	elpr 4609	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅, 1o} ↔ (𝑦 = ∅ ∨ 𝑦 = 1o))
10		elun 4155	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 ∪ 𝐵) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵))
11		velsn 4600	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
12	11	biimpri 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → 𝑦 ∈ {∅})
13	12	anim1i 627	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = ∅ ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
14	13	ancoms 464	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
15		opelxp 5694	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ↔ (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
16	14, 15	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴))
17	16	orcd 887	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
18		elun 4155	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
19	17, 18	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
20	19	orcd 887	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
21	20	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
22	12	anim1i 627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = ∅ ∧ 𝑧 ∈ 𝐵) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
23	22	ancoms 464	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
24		opelxp 5694	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ↔ (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
25	23, 24	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵))
26	25	orcd 887	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
27	26	olcd 888	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
28	27	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
29	21, 28	jaoi 871	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
30	29	com12 33	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
31		velsn 4600	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {1o} ↔ 𝑦 = 1o)
32	31	biimpri 231	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1o → 𝑦 ∈ {1o})
33	32	anim1i 627	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = 1o ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
34	33	ancoms 464	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
35		opelxp 5694	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴) ↔ (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
36	34, 35	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))
37	36	olcd 888	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
38	37	olcd 888	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
39	38	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
40	32	anim1i 627	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 1o ∧ 𝑧 ∈ 𝐵) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
41	40	ancoms 464	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
42		opelxp 5694	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵) ↔ (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
43	41, 42	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵))
44	43	olcd 888	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
45	44, 18	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
46	45	orcd 887	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
47	46	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
48	39, 47	jaoi 871	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
49	48	com12 33	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
50	30, 49	jaoi 871	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∨ 𝑦 = 1o) → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
51	50	imp 412	. . . . . . . . 9 ⊢ (((𝑦 = ∅ ∨ 𝑦 = 1o) ∧ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
52	9, 10, 51	syl2anb 610	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
53		elun 4155	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴)))
54		df-dju 10019	. . . . . . . . . . 11 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
55	54	eleq2i 2856	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
56		df-dju 10019	. . . . . . . . . . . 12 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴))
57	56	eleq2i 2856	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)))
58		elun 4155	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
59	57, 58	bitri 278	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
60	55, 59	orbi12i 928	. . . . . . . . 9 ⊢ ((⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
61	53, 60	bitri 278	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
62	52, 61	sylibr 237	. . . . . . 7 ⊢ ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵)) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
63	62	adantl 487	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
64		eleq1 2852	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))))
65	64	adantr 486	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → (𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))))
66	63, 65	mpbird 260	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
67	66	exlimivv 1967	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
68	7, 67	syl 18	. . 3 ⊢ (𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
69	68	ssriv 3917	. 2 ⊢ ({∅, 1o} × (𝐴 ∪ 𝐵)) ⊆ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))
70	6, 69	eqssi 3929	1 ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ∪ cun 4152  ∅c0 4281  {csn 4588  {cpr 4590  ⟨cop 4840   × cxp 5656  1_oc1o 8507   ⊔ cdju 10016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-suc 6394  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035  df-2nd 8036  df-1o 8514  df-dju 10019  df-inl 10020  df-inr 10021

This theorem is used by: (None)