Theorem oarec 7642

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 6658	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 ∅))
2		mpteq1 4737	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +𝑜 𝑥)))
3		mpt0 6021	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ (𝐴 +𝑜 𝑥)) = ∅
4	2, 3	syl6eq 2672	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ∅)
5	4	rneqd 5353	. . . . . 6 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran ∅)
6		rn0 5377	. . . . . 6 ⊢ ran ∅ = ∅
7	5, 6	syl6eq 2672	. . . . 5 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ∅)
8	7	uneq2d 3767	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ∅))
9	1, 8	eqeq12d 2637	. . 3 ⊢ (𝑧 = ∅ → ((𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) ↔ (𝐴 +𝑜 ∅) = (𝐴 ∪ ∅)))
10		oveq2 6658	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤))
11		mpteq1 4737	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
12	11	rneqd 5353	. . . . 5 ⊢ (𝑧 = 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
13	12	uneq2d 3767	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))))
14	10, 13	eqeq12d 2637	. . 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 6658	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 suc 𝑤))
16		mpteq1 4737	. . . . . 6 ⊢ (𝑧 = suc 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
17	16	rneqd 5353	. . . . 5 ⊢ (𝑧 = suc 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
18	17	uneq2d 3767	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))))
19	15, 18	eqeq12d 2637	. . 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 6658	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝐵))
21		mpteq1 4737	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
22	21	rneqd 5353	. . . . 5 ⊢ (𝑧 = 𝐵 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
23	22	uneq2d 3767	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
24	20, 23	eqeq12d 2637	. . 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 7596	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
26		un0 3967	. . . 4 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	syl6eqr 2674	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = (𝐴 ∪ ∅))
28		uneq1 3760	. . . . . 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 3770	. . . . . . 7 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
30		rexun 3793	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +𝑜 𝑥) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥)))
31		df-suc 5729	. . . . . . . . . . . 12 ⊢ suc 𝑤 = (𝑤 ∪ {𝑤})
32	31	rexeqi 3143	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +𝑜 𝑥))
33		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))
35	34	elrnmpt 5372	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥)))
36	33, 35	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥))
37		velsn 4193	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {(𝐴 +𝑜 𝑤)} ↔ 𝑦 = (𝐴 +𝑜 𝑤))
38		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
39		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
40	39	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑦 = (𝐴 +𝑜 𝑥) ↔ 𝑦 = (𝐴 +𝑜 𝑤)))
41	38, 40	rexsn 4223	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥) ↔ 𝑦 = (𝐴 +𝑜 𝑤))
42	37, 41	bitr4i 267	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {(𝐴 +𝑜 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥))
43	36, 42	orbi12i 543	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥)))
44	30, 32, 43	3bitr4i 292	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}))
45		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))
46		ovex 6678	. . . . . . . . . . 11 ⊢ (𝐴 +𝑜 𝑥) ∈ V
47	45, 46	elrnmpti 5376	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥))
48		elun 3753	. . . . . . . . . 10 ⊢ (𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}))
49	44, 47, 48	3bitr4i 292	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ 𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
50	49	eqriv 2619	. . . . . . . 8 ⊢ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)})
51	50	uneq2i 3764	. . . . . . 7 ⊢ (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
52	29, 51	eqtr4i 2647	. . . . . 6 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
53	28, 52	syl6eq 2672	. . . . 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))))
54		oasuc 7604	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +𝑜 suc 𝑤) = suc (𝐴 +𝑜 𝑤))
55		df-suc 5729	. . . . . . 7 ⊢ suc (𝐴 +𝑜 𝑤) = ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)})
56	54, 55	syl6eq 2672	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +𝑜 suc 𝑤) = ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)}))
57	56	eqeq1d 2624	. . . . 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)))))
58	53, 57	syl5ibr 236	. . . 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)))))
59	58	expcom 451	. . 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))))))
60		vex 3203	. . . . . . . 8 ⊢ 𝑧 ∈ V
61		oalim 7612	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +𝑜 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +𝑜 𝑤))
62	60, 61	mpanr1 719	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +𝑜 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +𝑜 𝑤))
63	62	ancoms 469	. . . . . 6 ⊢ ((Lim 𝑧 ∧ 𝐴 ∈ On) → (𝐴 +𝑜 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +𝑜 𝑤))
64	63	adantr 481	. . . . 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))
65		iuneq2 4537	. . . . . 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))))
66	65	adantl 482	. . . . 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))))
67		iunun 4604	. . . . . . 7 ⊢ ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
68		0ellim 5787	. . . . . . . . 9 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
69		ne0i 3921	. . . . . . . . 9 ⊢ (∅ ∈ 𝑧 → 𝑧 ≠ ∅)
70		iunconst 4529	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
71	68, 69, 70	3syl 18	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
72		limuni 5785	. . . . . . . . . . . 12 ⊢ (Lim 𝑧 → 𝑧 = ∪ 𝑧)
73	72	rexeqdv 3145	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥)))
74		df-rex 2918	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
75	36, 74	bitri 264	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
76	75	rexbii 3041	. . . . . . . . . . . 12 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
77		eluni2 4440	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤)
78	77	anbi1i 731	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
79		r19.41v 3089	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
80	78, 79	bitr4i 267	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
81	80	exbii 1774	. . . . . . . . . . . . 13 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
82		df-rex 2918	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
83		rexcom4 3225	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
84	81, 82, 83	3bitr4i 292	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
85	76, 84	bitr4i 267	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥))
86	73, 85	syl6rbbr 279	. . . . . . . . . 10 ⊢ (Lim 𝑧 → (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +𝑜 𝑥)))
87		eliun 4524	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
88		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))
89	88, 46	elrnmpti 5376	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +𝑜 𝑥))
90	86, 87, 89	3bitr4g 303	. . . . . . . . 9 ⊢ (Lim 𝑧 → (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ 𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))))
91	90	eqrdv 2620	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)))
92	71, 91	uneq12d 3768	. . . . . . 7 ⊢ (Lim 𝑧 → (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))))
93	67, 92	syl5eq 2668	. . . . . 6 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))))
94	93	ad2antrr 762	. . . . 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))))
95	64, 66, 94	3eqtrd 2660	. . . 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))))
96	95	exp31 630	. . 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))))))
97	9, 14, 19, 24, 27, 59, 96	tfinds3 7064	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))))
98	97	impcom 446	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   ↦ cmpt 4729  ran crn 5115  Oncon0 5723  Lim wlim 5724  suc csuc 5725  (class class class)co 6650   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  oacomf1o  7645  onacda  9019