Theorem djuunxp 9990

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 9989	. . 3 ⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
2		djuss 9989	. . . 4 ⊢ (𝐵 ⊔ 𝐴) ⊆ ({∅, 1o} × (𝐵 ∪ 𝐴))
3		uncom 4181	. . . . 5 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)
4	3	xpeq2i 5727	. . . 4 ⊢ ({∅, 1o} × (𝐴 ∪ 𝐵)) = ({∅, 1o} × (𝐵 ∪ 𝐴))
5	2, 4	sseqtrri 4046	. . 3 ⊢ (𝐵 ⊔ 𝐴) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
6	1, 5	unssi 4214	. 2 ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
7		elxpi 5722	. . . 4 ⊢ (𝑥 ∈ ({∅, 1o} × (𝐴 ∪ 𝐵)) → ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))))
8		vex 3492	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	8	elpr 4672	. . . . . . . . 9 ⊢ (𝑦 ∈ {∅, 1o} ↔ (𝑦 = ∅ ∨ 𝑦 = 1o))
10		elun 4176	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 ∪ 𝐵) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵))
11		velsn 4664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
12	11	biimpri 228	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → 𝑦 ∈ {∅})
13	12	anim1i 614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = ∅ ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
14	13	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
15		opelxp 5736	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ↔ (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐴))
16	14, 15	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴))
17	16	orcd 872	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
18		elun 4176	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
19	17, 18	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
20	19	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
21	20	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
22	12	anim1i 614	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = ∅ ∧ 𝑧 ∈ 𝐵) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
23	22	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
24		opelxp 5736	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ↔ (𝑦 ∈ {∅} ∧ 𝑧 ∈ 𝐵))
25	23, 24	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵))
26	25	orcd 872	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
27	26	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
28	27	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
29	21, 28	jaoi 856	. . . . . . . . . . . 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 4664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ {1o} ↔ 𝑦 = 1o)
32	31	biimpri 228	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 1o → 𝑦 ∈ {1o})
33	32	anim1i 614	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = 1o ∧ 𝑧 ∈ 𝐴) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
34	33	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
35		opelxp 5736	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴) ↔ (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐴))
36	34, 35	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))
37	36	olcd 873	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
38	37	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
39	38	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
40	32	anim1i 614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 = 1o ∧ 𝑧 ∈ 𝐵) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
41	40	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
42		opelxp 5736	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵) ↔ (𝑦 ∈ {1o} ∧ 𝑧 ∈ 𝐵))
43	41, 42	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵))
44	43	olcd 873	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
45	44, 18	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
46	45	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
47	46	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐵 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
48	39, 47	jaoi 856	. . . . . . . . . . . 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 856	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∨ 𝑦 = 1o) → ((𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
51	50	imp 406	. . . . . . . . 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 597	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
53		elun 4176	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴)))
54		df-dju 9970	. . . . . . . . . . 11 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
55	54	eleq2i 2836	. . . . . . . . . 10 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ⊔ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
56		df-dju 9970	. . . . . . . . . . . 12 ⊢ (𝐵 ⊔ 𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴))
57	56	eleq2i 2836	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 ⊔ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)))
58		elun 4176	. . . . . . . . . . 11 ⊢ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
59	57, 58	bitri 275	. . . . . . . . . 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 275	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
62	52, 61	sylibr 234	. . . . . . 7 ⊢ ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵)) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
63	62	adantl 481	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
64		eleq1 2832	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))))
65	64	adantr 480	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → (𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))))
66	63, 65	mpbird 257	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴 ∪ 𝐵))) → 𝑥 ∈ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)))
67	66	exlimivv 1931	. . . 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 4012	. 2 ⊢ ({∅, 1o} × (𝐴 ∪ 𝐵)) ⊆ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴))
70	6, 69	eqssi 4025	1 ⊢ ((𝐴 ⊔ 𝐵) ∪ (𝐵 ⊔ 𝐴)) = ({∅, 1o} × (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ∪ cun 3974  ∅c0 4352  {csn 4648  {cpr 4650  ⟨cop 4654   × cxp 5698  1_oc1o 8515   ⊔ cdju 9967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-suc 6401  df-iota 6525  df-fun 6575  df-fv 6581  df-1st 8030  df-2nd 8031  df-1o 8522  df-dju 9970  df-inl 9971  df-inr 9972

This theorem is referenced by: (None)