Theorem oarec 8526

Description: Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)

Assertion

Ref	Expression
oarec	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oarec

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

Step	Hyp	Ref	Expression
1		oveq2 7395	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 +o 𝑧) = (𝐴 +o ∅))
2		mpteq1 5196	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)))
3		mpt0 6660	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)) = ∅
4	2, 3	eqtrdi 2780	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
5	4	rneqd 5902	. . . . . 6 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran ∅)
6		rn0 5889	. . . . . 6 ⊢ ran ∅ = ∅
7	5, 6	eqtrdi 2780	. . . . 5 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
8	7	uneq2d 4131	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ∅))
9	1, 8	eqeq12d 2745	. . 3 ⊢ (𝑧 = ∅ → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o ∅) = (𝐴 ∪ ∅)))
10		oveq2 7395	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o 𝑤))
11		mpteq1 5196	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
12	11	rneqd 5902	. . . . 5 ⊢ (𝑧 = 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
13	12	uneq2d 4131	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))))
14	10, 13	eqeq12d 2745	. . 3 ⊢ (𝑧 = 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))))
15		oveq2 7395	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑤))
16		mpteq1 5196	. . . . . 6 ⊢ (𝑧 = suc 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
17	16	rneqd 5902	. . . . 5 ⊢ (𝑧 = suc 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
18	17	uneq2d 4131	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
19	15, 18	eqeq12d 2745	. . 3 ⊢ (𝑧 = suc 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
20		oveq2 7395	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 +o 𝑧) = (𝐴 +o 𝐵))
21		mpteq1 5196	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
22	21	rneqd 5902	. . . . 5 ⊢ (𝑧 = 𝐵 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
23	22	uneq2d 4131	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
24	20, 23	eqeq12d 2745	. . 3 ⊢ (𝑧 = 𝐵 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))))
25		oa0 8480	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
26		un0 4357	. . . 4 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	eqtr4di 2782	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 ∪ ∅))
28		uneq1 4124	. . . . . 6 ⊢ ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}))
29		unass 4135	. . . . . . 7 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
30		rexun 4159	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
31		df-suc 6338	. . . . . . . . . . . 12 ⊢ suc 𝑤 = (𝑤 ∪ {𝑤})
32	31	rexeqi 3298	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥))
33		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))
34	33	elrnmpt 5922	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥)))
35	34	elv 3452	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥))
36		velsn 4605	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ 𝑦 = (𝐴 +o 𝑤))
37		vex 3451	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
38		oveq2 7395	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
39	38	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤)))
40	37, 39	rexsn 4646	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤))
41	36, 40	bitr4i 278	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥))
42	35, 41	orbi12i 914	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
43	30, 32, 42	3bitr4i 303	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
44		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))
45		ovex 7420	. . . . . . . . . . 11 ⊢ (𝐴 +o 𝑥) ∈ V
46	44, 45	elrnmpti 5926	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥))
47		elun 4116	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
48	43, 46, 47	3bitr4i 303	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
49	48	eqriv 2726	. . . . . . . 8 ⊢ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})
50	49	uneq2i 4128	. . . . . . 7 ⊢ (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
51	29, 50	eqtr4i 2755	. . . . . 6 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
52	28, 51	eqtrdi 2780	. . . . 5 ⊢ ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
53		oasuc 8488	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = suc (𝐴 +o 𝑤))
54		df-suc 6338	. . . . . . 7 ⊢ suc (𝐴 +o 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)})
55	53, 54	eqtrdi 2780	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}))
56	55	eqeq1d 2731	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) ↔ ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
57	52, 56	imbitrrid 246	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
58	57	expcom 413	. . 3 ⊢ (𝑤 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))))
59		vex 3451	. . . . . . . 8 ⊢ 𝑧 ∈ V
60		oalim 8496	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
61	59, 60	mpanr1 703	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
62	61	ancoms 458	. . . . . 6 ⊢ ((Lim 𝑧 ∧ 𝐴 ∈ On) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
63	62	adantr 480	. . . . 5 ⊢ (((Lim 𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
64		iuneq2 4975	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))))
65	64	adantl 481	. . . . 5 ⊢ (((Lim 𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))))
66		iunun 5057	. . . . . . 7 ⊢ ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
67		0ellim 6396	. . . . . . . . 9 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
68		ne0i 4304	. . . . . . . . 9 ⊢ (∅ ∈ 𝑧 → 𝑧 ≠ ∅)
69		iunconst 4965	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
70	67, 68, 69	3syl 18	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
71		df-rex 3054	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
72	35, 71	bitri 275	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
73	72	rexbii 3076	. . . . . . . . . . . 12 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
74		eluni2 4875	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤)
75	74	anbi1i 624	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
76		r19.41v 3167	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
77	75, 76	bitr4i 278	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
78	77	exbii 1848	. . . . . . . . . . . . 13 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
79		df-rex 3054	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)))
80		rexcom4 3264	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
81	78, 79, 80	3bitr4i 303	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
82	73, 81	bitr4i 278	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥))
83		limuni 6394	. . . . . . . . . . . 12 ⊢ (Lim 𝑧 → 𝑧 = ∪ 𝑧)
84	83	rexeqdv 3300	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥)))
85	82, 84	bitr4id 290	. . . . . . . . . 10 ⊢ (Lim 𝑧 → (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥)))
86		eliun 4959	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
87		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))
88	87, 45	elrnmpti 5926	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥))
89	85, 86, 88	3bitr4g 314	. . . . . . . . 9 ⊢ (Lim 𝑧 → (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
90	89	eqrdv 2727	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))
91	70, 90	uneq12d 4132	. . . . . . 7 ⊢ (Lim 𝑧 → (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
92	66, 91	eqtrid 2776	. . . . . 6 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
93	92	ad2antrr 726	. . . . 5 ⊢ (((Lim 𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
94	63, 65, 93	3eqtrd 2768	. . . 4 ⊢ (((Lim 𝑧 ∧ 𝐴 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
95	94	exp31 419	. . 3 ⊢ (Lim 𝑧 → (𝐴 ∈ On → (∀𝑤 ∈ 𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))))
96	9, 14, 19, 24, 27, 58, 95	tfinds3 7841	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))))
97	96	impcom 407	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∪ cun 3912  ∅c0 4296  {csn 4589  ∪ cuni 4871  ∪ ciun 4955   ↦ cmpt 5188  ran crn 5639  Oncon0 6332  Lim wlim 6333  suc csuc 6334  (class class class)co 7387   +_o coa 8431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-oadd 8438

This theorem is referenced by:  oacomf1o  8529  onadju  10147