Theorem oawordeulem 7521

Description: Lemma for oawordex 7524. (Contributed by NM, 11-Dec-2004.)

Hypotheses

Ref	Expression
oawordeulem.1	⊢ 𝐴 ∈ On
oawordeulem.2	⊢ 𝐵 ∈ On
oawordeulem.3	⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}

Assertion

Ref	Expression
oawordeulem	⊢ (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)

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

Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem oawordeulem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oawordeulem.3	. . . . . 6 ⊢ 𝑆 = {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
2		ssrab2 3650	. . . . . 6 ⊢ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
3	1, 2	eqsstri 3598	. . . . 5 ⊢ 𝑆 ⊆ On
4		oawordeulem.2	. . . . . . 7 ⊢ 𝐵 ∈ On
5		oawordeulem.1	. . . . . . . 8 ⊢ 𝐴 ∈ On
6		oaword2 7520	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
7	4, 5, 6	mp2an 704	. . . . . . 7 ⊢ 𝐵 ⊆ (𝐴 +𝑜 𝐵)
8		oveq2 6557	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
9	8	sseq2d 3596	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
10	9, 1	elrab2 3333	. . . . . . 7 ⊢ (𝐵 ∈ 𝑆 ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
11	4, 7, 10	mpbir2an 957	. . . . . 6 ⊢ 𝐵 ∈ 𝑆
12	11	ne0ii 3882	. . . . 5 ⊢ 𝑆 ≠ ∅
13		oninton 6892	. . . . 5 ⊢ ((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ On)
14	3, 12, 13	mp2an 704	. . . 4 ⊢ ∩ 𝑆 ∈ On
15		onzsl 6938	. . . . . . . 8 ⊢ (∩ 𝑆 ∈ On ↔ (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)))
16	14, 15	mpbi 219	. . . . . . 7 ⊢ (∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆))
17		oveq2 6557	. . . . . . . . . . 11 ⊢ (∩ 𝑆 = ∅ → (𝐴 +𝑜 ∩ 𝑆) = (𝐴 +𝑜 ∅))
18		oa0 7483	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
19	5, 18	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐴 +𝑜 ∅) = 𝐴
20	17, 19	syl6eq 2660	. . . . . . . . . 10 ⊢ (∩ 𝑆 = ∅ → (𝐴 +𝑜 ∩ 𝑆) = 𝐴)
21	20	sseq1d 3595	. . . . . . . . 9 ⊢ (∩ 𝑆 = ∅ → ((𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
22	21	biimprd 237	. . . . . . . 8 ⊢ (∩ 𝑆 = ∅ → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵))
23		oveq2 6557	. . . . . . . . . . . 12 ⊢ (∩ 𝑆 = suc 𝑧 → (𝐴 +𝑜 ∩ 𝑆) = (𝐴 +𝑜 suc 𝑧))
24		oasuc 7491	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
25	5, 24	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ On → (𝐴 +𝑜 suc 𝑧) = suc (𝐴 +𝑜 𝑧))
26	23, 25	sylan9eqr 2666	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ ∩ 𝑆 = suc 𝑧) → (𝐴 +𝑜 ∩ 𝑆) = suc (𝐴 +𝑜 𝑧))
27		vex 3176	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
28	27	sucid 5721	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ suc 𝑧
29		eleq2 2677	. . . . . . . . . . . . . 14 ⊢ (∩ 𝑆 = suc 𝑧 → (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ suc 𝑧))
30	28, 29	mpbiri 247	. . . . . . . . . . . . 13 ⊢ (∩ 𝑆 = suc 𝑧 → 𝑧 ∈ ∩ 𝑆)
31	14	oneli 5752	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∩ 𝑆 → 𝑧 ∈ On)
32	1	inteqi 4414	. . . . . . . . . . . . . . . . 17 ⊢ ∩ 𝑆 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
33	32	eleq2i 2680	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ∩ 𝑆 ↔ 𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
34		oveq2 6557	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑧))
35	34	sseq2d 3596	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
36	35	onnminsb 6896	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
37	33, 36	syl5bi 231	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ 𝑆 → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
38		oacl 7502	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ On ∧ 𝑧 ∈ On) → (𝐴 +𝑜 𝑧) ∈ On)
39	5, 38	mpan 702	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ On → (𝐴 +𝑜 𝑧) ∈ On)
40		ontri1 5674	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ On ∧ (𝐴 +𝑜 𝑧) ∈ On) → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
41	4, 39, 40	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ On → (𝐵 ⊆ (𝐴 +𝑜 𝑧) ↔ ¬ (𝐴 +𝑜 𝑧) ∈ 𝐵))
42	41	con2bid 343	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ On → ((𝐴 +𝑜 𝑧) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑧)))
43	37, 42	sylibrd 248	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑧 ∈ ∩ 𝑆 → (𝐴 +𝑜 𝑧) ∈ 𝐵))
44	31, 43	mpcom 37	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∩ 𝑆 → (𝐴 +𝑜 𝑧) ∈ 𝐵)
45	4	onordi 5749	. . . . . . . . . . . . . 14 ⊢ Ord 𝐵
46		ordsucss 6910	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐵 → ((𝐴 +𝑜 𝑧) ∈ 𝐵 → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵))
47	45, 46	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((𝐴 +𝑜 𝑧) ∈ 𝐵 → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
48	30, 44, 47	3syl 18	. . . . . . . . . . . 12 ⊢ (∩ 𝑆 = suc 𝑧 → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ ∩ 𝑆 = suc 𝑧) → suc (𝐴 +𝑜 𝑧) ⊆ 𝐵)
50	26, 49	eqsstrd 3602	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ ∩ 𝑆 = suc 𝑧) → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵)
51	50	rexlimiva 3010	. . . . . . . . 9 ⊢ (∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵)
52	51	a1d 25	. . . . . . . 8 ⊢ (∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵))
53		oalim 7499	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐴 +𝑜 ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐴 +𝑜 𝑧))
54	5, 53	mpan 702	. . . . . . . . . 10 ⊢ ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐴 +𝑜 ∩ 𝑆) = ∪ 𝑧 ∈ ∩ 𝑆(𝐴 +𝑜 𝑧))
55		iunss 4497	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ ∩ 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵 ↔ ∀𝑧 ∈ ∩ 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵)
56	4	onelssi 5753	. . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝑧) ∈ 𝐵 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
57	44, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∩ 𝑆 → (𝐴 +𝑜 𝑧) ⊆ 𝐵)
58	55, 57	mprgbir 2911	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ∩ 𝑆(𝐴 +𝑜 𝑧) ⊆ 𝐵
59	54, 58	syl6eqss 3618	. . . . . . . . 9 ⊢ ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵)
60	59	a1d 25	. . . . . . . 8 ⊢ ((∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵))
61	22, 52, 60	3jaoi 1383	. . . . . . 7 ⊢ ((∩ 𝑆 = ∅ ∨ ∃𝑧 ∈ On ∩ 𝑆 = suc 𝑧 ∨ (∩ 𝑆 ∈ V ∧ Lim ∩ 𝑆)) → (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵))
62	16, 61	ax-mp 5	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵)
63	9	rspcev 3282	. . . . . . . . 9 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)) → ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦))
64	4, 7, 63	mp2an 704	. . . . . . . 8 ⊢ ∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦)
65		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵
66		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐴
67		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 +𝑜
68		nfrab1 3099	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
69	68	nfint 4421	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
70	66, 67, 69	nfov 6575	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
71	65, 70	nfss 3561	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
72		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
73	72	sseq2d 3596	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
74	71, 73	onminsb 6891	. . . . . . . 8 ⊢ (∃𝑦 ∈ On 𝐵 ⊆ (𝐴 +𝑜 𝑦) → 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
75	64, 74	ax-mp 5	. . . . . . 7 ⊢ 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
76	32	oveq2i 6560	. . . . . . 7 ⊢ (𝐴 +𝑜 ∩ 𝑆) = (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
77	75, 76	sseqtr4i 3601	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 +𝑜 ∩ 𝑆)
78	62, 77	jctir 559	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 +𝑜 ∩ 𝑆)))
79		eqss 3583	. . . . 5 ⊢ ((𝐴 +𝑜 ∩ 𝑆) = 𝐵 ↔ ((𝐴 +𝑜 ∩ 𝑆) ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 +𝑜 ∩ 𝑆)))
80	78, 79	sylibr 223	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 +𝑜 ∩ 𝑆) = 𝐵)
81		oveq2 6557	. . . . . 6 ⊢ (𝑥 = ∩ 𝑆 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∩ 𝑆))
82	81	eqeq1d 2612	. . . . 5 ⊢ (𝑥 = ∩ 𝑆 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 ∩ 𝑆) = 𝐵))
83	82	rspcev 3282	. . . 4 ⊢ ((∩ 𝑆 ∈ On ∧ (𝐴 +𝑜 ∩ 𝑆) = 𝐵) → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
84	14, 80, 83	sylancr 694	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
85		eqtr3 2631	. . . . 5 ⊢ (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
86		oacan 7515	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
87	5, 86	mp3an1 1403	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦) ↔ 𝑥 = 𝑦))
88	85, 87	syl5ib 233	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦))
89	88	rgen2a 2960	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)
90	84, 89	jctir 559	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)))
91		oveq2 6557	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑦))
92	91	eqeq1d 2612	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 𝑦) = 𝐵))
93	92	reu4 3367	. 2 ⊢ (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ↔ (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (((𝐴 +𝑜 𝑥) = 𝐵 ∧ (𝐴 +𝑜 𝑦) = 𝐵) → 𝑥 = 𝑦)))
94	90, 93	sylibr 223	1 ⊢ (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  ∪ ciun 4455  Ord word 5639  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549   +_𝑜 coa 7444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  oawordeu  7522