Theorem djuunxp 9857

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 9856	. . 3 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		djuss 9856	. . . 4 ⊢ (𝐵 ⊔ 𝐴) ⊆ ({∅, 1o} × (𝐵 ∪ 𝐴))
3		uncom 4113	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
4	3	xpeq2i 5660	. . . 4 ⊢ ({∅, 1o} × (𝐴 ∪ 𝐵)) = ({∅, 1o} × (𝐵 ∪ 𝐴))
5	2, 4	sseqtrri 3981	. . 3 ⊢ (𝐵 ⊔ 𝐴) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
6	1, 5	unssi 4145	. 2 ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
7		elxpi 5655	. . . 4 ⊢ (𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))))
8		vex 3449	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	elpr 4609	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅, 1o} ↔ (𝑦 = ∅ ∨ 𝑦 = 1o))
10		elun 4108	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 ∪ 𝐵) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵))
11		velsn 4602	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
12	11	biimpri 227	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → 𝑦 ∈ {∅})
13	12	anim1i 615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = ∅ ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
14	13	ancoms 459	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
15		opelxp 5669	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ↔ (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
16	14, 15	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴))
17	16	orcd 871	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
18		elun 4108	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
19	17, 18	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
20	19	orcd 871	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
21	20	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
22	12	anim1i 615	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = ∅ ∧ 𝑧 ∈ 𝐵) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
23	22	ancoms 459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
24		opelxp 5669	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ↔ (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
25	23, 24	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵))
26	25	orcd 871	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
27	26	olcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
28	27	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
29	21, 28	jaoi 855	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
30	29	com12 32	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
31		velsn 4602	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {1o} ↔ 𝑦 = 1o)
32	31	biimpri 227	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1o → 𝑦 ∈ {1o})
33	32	anim1i 615	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = 1o ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
34	33	ancoms 459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
35		opelxp 5669	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴) ↔ (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
36	34, 35	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))
37	36	olcd 872	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
38	37	olcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
39	38	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
40	32	anim1i 615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 1o ∧ 𝑧 ∈ 𝐵) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
41	40	ancoms 459	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
42		opelxp 5669	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵) ↔ (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
43	41, 42	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵))
44	43	olcd 872	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
45	44, 18	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
46	45	orcd 871	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
47	46	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
48	39, 47	jaoi 855	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
49	48	com12 32	. . . . . . . . . . 11 ⊢ (𝑦 = 1o → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
50	30, 49	jaoi 855	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∨ 𝑦 = 1o) → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
51	50	imp 407	. . . . . . . . 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 598	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
53		elun 4108	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴)))
54		df-dju 9837	. . . . . . . . . . 11 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
55	54	eleq2i 2829	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
56		df-dju 9837	. . . . . . . . . . . 12 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴))
57	56	eleq2i 2829	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)))
58		elun 4108	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
59	57, 58	bitri 274	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
60	55, 59	orbi12i 913	. . . . . . . . 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 274	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
62	52, 61	sylibr 233	. . . . . . 7 ⊢ ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵)) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
63	62	adantl 482	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
64		eleq1 2825	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))))
65	64	adantr 481	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → (𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))))
66	63, 65	mpbird 256	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
67	66	exlimivv 1935	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
68	7, 67	syl 17	. . 3 ⊢ (𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
69	68	ssriv 3948	. 2 ⊢ ({∅, 1o} × (𝐴 ∪ 𝐵)) ⊆ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))
70	6, 69	eqssi 3960	1 ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ∪ cun 3908  ∅c0 4282  {csn 4586  {cpr 4588  ⟨cop 4592   × cxp 5631  1_oc1o 8405   ⊔ cdju 9834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-suc 6323  df-iota 6448  df-fun 6498  df-fv 6504  df-1st 7921  df-2nd 7922  df-1o 8412  df-dju 9837  df-inl 9838  df-inr 9839

This theorem is referenced by: (None)