Theorem oarec 7529

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 6557	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 ∅))
2		mpteq1 4665	. . . . . . . 8 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +𝑜 𝑥)))
3		mpt0 5934	. . . . . . . 8 ⊢ (𝑥 ∈ ∅ ↦ (𝐴 +𝑜 𝑥)) = ∅
4	2, 3	syl6eq 2660	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ∅)
5	4	rneqd 5274	. . . . . 6 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran ∅)
6		rn0 5298	. . . . . 6 ⊢ ran ∅ = ∅
7	5, 6	syl6eq 2660	. . . . 5 ⊢ (𝑧 = ∅ → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ∅)
8	7	uneq2d 3729	. . . 4 ⊢ (𝑧 = ∅ → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ∅))
9	1, 8	eqeq12d 2625	. . 3 ⊢ (𝑧 = ∅ → ((𝐴 +𝑜 𝑧) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) ↔ (𝐴 +𝑜 ∅) = (𝐴 ∪ ∅)))
10		oveq2 6557	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝑤))
11		mpteq1 4665	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
12	11	rneqd 5274	. . . . 5 ⊢ (𝑧 = 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
13	12	uneq2d 3729	. . . 4 ⊢ (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))))
14	10, 13	eqeq12d 2625	. . 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 6557	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 suc 𝑤))
16		mpteq1 4665	. . . . . 6 ⊢ (𝑧 = suc 𝑤 → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
17	16	rneqd 5274	. . . . 5 ⊢ (𝑧 = suc 𝑤 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
18	17	uneq2d 3729	. . . 4 ⊢ (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))))
19	15, 18	eqeq12d 2625	. . 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 6557	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 +𝑜 𝑧) = (𝐴 +𝑜 𝐵))
21		mpteq1 4665	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
22	21	rneqd 5274	. . . . 5 ⊢ (𝑧 = 𝐵 → ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
23	22	uneq2d 3729	. . . 4 ⊢ (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
24	20, 23	eqeq12d 2625	. . 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 7483	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = 𝐴)
26		un0 3919	. . . 4 ⊢ (𝐴 ∪ ∅) = 𝐴
27	25, 26	syl6eqr 2662	. . 3 ⊢ (𝐴 ∈ On → (𝐴 +𝑜 ∅) = (𝐴 ∪ ∅))
28		uneq1 3722	. . . . . 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 3732	. . . . . . 7 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
30		rexun 3755	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +𝑜 𝑥) ↔ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥)))
31		df-suc 5646	. . . . . . . . . . . 12 ⊢ suc 𝑤 = (𝑤 ∪ {𝑤})
32	31	rexeqi 3120	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +𝑜 𝑥))
33		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))
35	34	elrnmpt 5293	. . . . . . . . . . . . 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 4141	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {(𝐴 +𝑜 𝑤)} ↔ 𝑦 = (𝐴 +𝑜 𝑤))
38		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
39		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 𝑤))
40	39	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑦 = (𝐴 +𝑜 𝑥) ↔ 𝑦 = (𝐴 +𝑜 𝑤)))
41	38, 40	rexsn 4170	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥) ↔ 𝑦 = (𝐴 +𝑜 𝑤))
42	37, 41	bitr4i 266	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {(𝐴 +𝑜 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +𝑜 𝑥))
43	36, 42	orbi12i 542	. . . . . . . . . . 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 291	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥) ↔ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∨ 𝑦 ∈ {(𝐴 +𝑜 𝑤)}))
45		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))
46		ovex 6577	. . . . . . . . . . 11 ⊢ (𝐴 +𝑜 𝑥) ∈ V
47	45, 46	elrnmpti 5297	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +𝑜 𝑥))
48		elun 3715	. . . . . . . . . 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 291	. . . . . . . . 9 ⊢ (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ 𝑦 ∈ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
50	49	eqriv 2607	. . . . . . . 8 ⊢ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)) = (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)})
51	50	uneq2i 3726	. . . . . . 7 ⊢ (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ (ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ∪ {(𝐴 +𝑜 𝑤)}))
52	29, 51	eqtr4i 2635	. . . . . 6 ⊢ ((𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) ∪ {(𝐴 +𝑜 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +𝑜 𝑥)))
53	28, 52	syl6eq 2660	. . . . 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 7491	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +𝑜 suc 𝑤) = suc (𝐴 +𝑜 𝑤))
55		df-suc 5646	. . . . . . 7 ⊢ suc (𝐴 +𝑜 𝑤) = ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)})
56	54, 55	syl6eq 2660	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +𝑜 suc 𝑤) = ((𝐴 +𝑜 𝑤) ∪ {(𝐴 +𝑜 𝑤)}))
57	56	eqeq1d 2612	. . . . 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 235	. . . 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 450	. . 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 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
61		oalim 7499	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +𝑜 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +𝑜 𝑤))
62	60, 61	mpanr1 715	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +𝑜 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +𝑜 𝑤))
63	62	ancoms 468	. . . . . 6 ⊢ ((Lim 𝑧 ∧ 𝐴 ∈ On) → (𝐴 +𝑜 𝑧) = ∪ 𝑤 ∈ 𝑧 (𝐴 +𝑜 𝑤))
64	63	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))
65		iuneq2 4473	. . . . . 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 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))))
67		iunun 4540	. . . . . . 7 ⊢ ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (∪ 𝑤 ∈ 𝑧 𝐴 ∪ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
68		0ellim 5704	. . . . . . . . 9 ⊢ (Lim 𝑧 → ∅ ∈ 𝑧)
69		ne0i 3880	. . . . . . . . 9 ⊢ (∅ ∈ 𝑧 → 𝑧 ≠ ∅)
70		iunconst 4465	. . . . . . . . 9 ⊢ (𝑧 ≠ ∅ → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
71	68, 69, 70	3syl 18	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 𝐴 = 𝐴)
72		limuni 5702	. . . . . . . . . . . 12 ⊢ (Lim 𝑧 → 𝑧 = ∪ 𝑧)
73	72	rexeqdv 3122	. . . . . . . . . . 11 ⊢ (Lim 𝑧 → (∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥)))
74		df-rex 2902	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝑤 𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
75	36, 74	bitri 263	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
76	75	rexbii 3023	. . . . . . . . . . . 12 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
77		eluni2 4376	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ∪ 𝑧 ↔ ∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤)
78	77	anbi1i 727	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
79		r19.41v 3070	. . . . . . . . . . . . . . 15 ⊢ (∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ (∃𝑤 ∈ 𝑧 𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
80	78, 79	bitr4i 266	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
81	80	exbii 1764	. . . . . . . . . . . . 13 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
82		df-rex 2902	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑥(𝑥 ∈ ∪ 𝑧 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
83		rexcom4 3198	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)) ↔ ∃𝑥∃𝑤 ∈ 𝑧 (𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
84	81, 82, 83	3bitr4i 291	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥) ↔ ∃𝑤 ∈ 𝑧 ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝑦 = (𝐴 +𝑜 𝑥)))
85	76, 84	bitr4i 266	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ ∪ 𝑧𝑦 = (𝐴 +𝑜 𝑥))
86	73, 85	syl6rbbr 278	. . . . . . . . . 10 ⊢ (Lim 𝑧 → (∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +𝑜 𝑥)))
87		eliun 4460	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑤 ∈ 𝑧 𝑦 ∈ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)))
88		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))
89	88, 46	elrnmpti 5297	. . . . . . . . . 10 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)) ↔ ∃𝑥 ∈ 𝑧 𝑦 = (𝐴 +𝑜 𝑥))
90	86, 87, 89	3bitr4g 302	. . . . . . . . 9 ⊢ (Lim 𝑧 → (𝑦 ∈ ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) ↔ 𝑦 ∈ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))))
91	90	eqrdv 2608	. . . . . . . 8 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥)) = ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥)))
92	71, 91	uneq12d 3730	. . . . . . 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 2656	. . . . . 6 ⊢ (Lim 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝐴 ∪ ran (𝑥 ∈ 𝑤 ↦ (𝐴 +𝑜 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ 𝑧 ↦ (𝐴 +𝑜 𝑥))))
94	93	ad2antrr 758	. . . . 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 2648	. . . 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 628	. . 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 6956	. 2 ⊢ (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))))
98	97	impcom 445	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ran crn 5039  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-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-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:  oacomf1o  7532  onacda  8902