Theorem oarec 8584

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 7438	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 +o 𝑧) = (𝐴 +o ∅))
2		mpteq1 5197	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)))
3		mpt0 6692	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)) = ∅
4	2, 3	eqtrdi 2814	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
5	4	rneqd 5925	. . . . . 6 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran ∅)
6		rn0 5913	. . . . . 6 ⊢ ran ∅ = ∅
7	5, 6	eqtrdi 2814	. . . . 5 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
8	7	uneq2d 4170	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ∅))
9	1, 8	eqeq12d 2779	. . 3 ⊢ (𝑧 = ∅ → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o ∅) = (𝐴 ∪ ∅)))
10		oveq2 7438	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o 𝑤))
11		mpteq1 5197	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
12	11	rneqd 5925	. . . . 5 ⊢ (𝑧 = 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
13	12	uneq2d 4170	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))))
14	10, 13	eqeq12d 2779	. . 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 7438	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑤))
16		mpteq1 5197	. . . . . 6 ⊢ (𝑧 = suc 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
17	16	rneqd 5925	. . . . 5 ⊢ (𝑧 = suc 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
18	17	uneq2d 4170	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
19	15, 18	eqeq12d 2779	. . 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 7438	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 +o 𝑧) = (𝐴 +o 𝐵))
21		mpteq1 5197	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
22	21	rneqd 5925	. . . . 5 ⊢ (𝑧 = 𝐵 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
23	22	uneq2d 4170	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
24	20, 23	eqeq12d 2779	. . 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 8538	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
26		un0 4347	. . . 4 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	eqtr4di 2816	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 ∪ ∅))
28		uneq1 4163	. . . . . 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 4173	. . . . . . 7 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
30		rexun 4197	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
31		df-suc 6375	. . . . . . . . . . . 12 ⊢ suc 𝑤 = (𝑤 ∪ {𝑤})
32	31	rexeqi 3322	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥))
33		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))
34	33	elrnmpt 5945	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥)))
35	34	elv 3459	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥))
36		velsn 4600	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ 𝑦 = (𝐴 +o 𝑤))
37		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
38		oveq2 7438	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
39	38	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤)))
40	37, 39	rexsn 4643	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤))
41	36, 40	bitr4i 281	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥))
42	35, 41	orbi12i 928	. . . . . . . . . . 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 306	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
44		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))
45		ovex 7463	. . . . . . . . . . 11 ⊢ (𝐴 +o 𝑥) ∈ V
46	44, 45	elrnmpti 5949	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥))
47		elun 4155	. . . . . . . . . 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 306	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
49	48	eqriv 2760	. . . . . . . 8 ⊢ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})
50	49	uneq2i 4167	. . . . . . 7 ⊢ (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
51	29, 50	eqtr4i 2789	. . . . . 6 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
52	28, 51	eqtrdi 2814	. . . . 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 8546	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = suc (𝐴 +o 𝑤))
54		df-suc 6375	. . . . . . 7 ⊢ suc (𝐴 +o 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)})
55	53, 54	eqtrdi 2814	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}))
56	55	eqeq1d 2765	. . . . 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 249	. . . 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 419	. . 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 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
60		oalim 8554	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
61	59, 60	mpanr1 716	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
62	61	ancoms 464	. . . . . 6 ⊢ ((Lim 𝑧 ∧ 𝐴 ∈ On) → (𝐴 +o 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +o 𝑤))
63	62	adantr 486	. . . . 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 4974	. . . . . 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 487	. . . . 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 5056	. . . . . . 7 ⊢ ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
67		0ellim 6434	. . . . . . . . 9 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
68		ne0i 4290	. . . . . . . . 9 ⊢ (∅ ∈ 𝑧 → 𝑧 ≠ ∅)
69		iunconst 4964	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
70	67, 68, 69	3syl 19	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
71		df-rex 3091	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
72	35, 71	bitri 278	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
73	72	rexbii 3113	. . . . . . . . . . . 12 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
74		eluni2 4874	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤)
75	74	anbi1i 636	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
76		r19.41v 3196	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
77	75, 76	bitr4i 281	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
78	77	exbii 1880	. . . . . . . . . . . . 13 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
79		df-rex 3091	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +o 𝑥)))
80		rexcom4 3293	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
81	78, 79, 80	3bitr4i 306	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +o 𝑥)))
82	73, 81	bitr4i 281	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥))
83		limuni 6432	. . . . . . . . . . . 12 ⊢ (Lim 𝑧 → 𝑧 = ∪ 𝑧)
84	83	rexeqdv 3324	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +o 𝑥)))
85	82, 84	bitr4id 293	. . . . . . . . . 10 ⊢ (Lim 𝑧 → (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥)))
86		eliun 4958	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)))
87		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))
88	87, 45	elrnmpti 5949	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +o 𝑥))
89	85, 86, 88	3bitr4g 317	. . . . . . . . 9 ⊢ (Lim 𝑧 → (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
90	89	eqrdv 2761	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥)))
91	70, 90	uneq12d 4171	. . . . . . 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 2810	. . . . . 6 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +o 𝑥))))
93	92	ad2antrr 739	. . . . 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 2802	. . . 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 425	. . 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 7886	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))))
97	96	impcom 413	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ∪ cun 4152  ∅c0 4281  {csn 4588  ∪ cuni 4870  ∪ ciun 4954   ↦ cmpt 5189  ran crn 5656  Oncon0 6369  Lim wlim 6370  suc csuc 6371  (class class class)co 7430   +_o coa 8487

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-oadd 8494

This theorem is used by:  oacomf1o  8587  onadju  10285