Theorem ituniiun 10178

Description: Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.)

Hypothesis

Ref	Expression
ituni.u	⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))

Assertion

Ref	Expression
ituniiun	⊢ (𝐴 ∈ 𝑉 → ((𝑈‘𝐴)‘suc 𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑎   𝑥,𝐵,𝑦,𝑎   𝑈,𝑎

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑎)

Proof of Theorem ituniiun

Dummy variables 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . 4 ⊢ (𝑏 = 𝐴 → (𝑈‘𝑏) = (𝑈‘𝐴))
2	1	fveq1d 6776	. . 3 ⊢ (𝑏 = 𝐴 → ((𝑈‘𝑏)‘suc 𝐵) = ((𝑈‘𝐴)‘suc 𝐵))
3		iuneq1 4940	. . 3 ⊢ (𝑏 = 𝐴 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵))
4	2, 3	eqeq12d 2754	. 2 ⊢ (𝑏 = 𝐴 → (((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵) ↔ ((𝑈‘𝐴)‘suc 𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵)))
5		suceq 6331	. . . . . 6 ⊢ (𝑑 = ∅ → suc 𝑑 = suc ∅)
6	5	fveq2d 6778	. . . . 5 ⊢ (𝑑 = ∅ → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc ∅))
7		fveq2 6774	. . . . . 6 ⊢ (𝑑 = ∅ → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘∅))
8	7	iuneq2d 4953	. . . . 5 ⊢ (𝑑 = ∅ → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅))
9	6, 8	eqeq12d 2754	. . . 4 ⊢ (𝑑 = ∅ → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc ∅) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅)))
10		suceq 6331	. . . . . 6 ⊢ (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
11	10	fveq2d 6778	. . . . 5 ⊢ (𝑑 = 𝑐 → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc 𝑐))
12		fveq2 6774	. . . . . 6 ⊢ (𝑑 = 𝑐 → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘𝑐))
13	12	iuneq2d 4953	. . . . 5 ⊢ (𝑑 = 𝑐 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐))
14	11, 13	eqeq12d 2754	. . . 4 ⊢ (𝑑 = 𝑐 → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐)))
15		suceq 6331	. . . . . 6 ⊢ (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
16	15	fveq2d 6778	. . . . 5 ⊢ (𝑑 = suc 𝑐 → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc suc 𝑐))
17		fveq2 6774	. . . . . 6 ⊢ (𝑑 = suc 𝑐 → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘suc 𝑐))
18	17	iuneq2d 4953	. . . . 5 ⊢ (𝑑 = suc 𝑐 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐))
19	16, 18	eqeq12d 2754	. . . 4 ⊢ (𝑑 = suc 𝑐 → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐)))
20		suceq 6331	. . . . . 6 ⊢ (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
21	20	fveq2d 6778	. . . . 5 ⊢ (𝑑 = 𝐵 → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc 𝐵))
22		fveq2 6774	. . . . . 6 ⊢ (𝑑 = 𝐵 → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘𝐵))
23	22	iuneq2d 4953	. . . . 5 ⊢ (𝑑 = 𝐵 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵))
24	21, 23	eqeq12d 2754	. . . 4 ⊢ (𝑑 = 𝐵 → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵)))
25		uniiun 4988	. . . . 5 ⊢ ∪ 𝑏 = ∪ 𝑎 ∈ 𝑏 𝑎
26		ituni.u	. . . . . . 7 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
27	26	itunisuc 10175	. . . . . 6 ⊢ ((𝑈‘𝑏)‘suc ∅) = ∪ ((𝑈‘𝑏)‘∅)
28	26	ituni0 10174	. . . . . . . 8 ⊢ (𝑏 ∈ V → ((𝑈‘𝑏)‘∅) = 𝑏)
29	28	elv 3438	. . . . . . 7 ⊢ ((𝑈‘𝑏)‘∅) = 𝑏
30	29	unieqi 4852	. . . . . 6 ⊢ ∪ ((𝑈‘𝑏)‘∅) = ∪ 𝑏
31	27, 30	eqtri 2766	. . . . 5 ⊢ ((𝑈‘𝑏)‘suc ∅) = ∪ 𝑏
32	26	ituni0 10174	. . . . . 6 ⊢ (𝑎 ∈ 𝑏 → ((𝑈‘𝑎)‘∅) = 𝑎)
33	32	iuneq2i 4945	. . . . 5 ⊢ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅) = ∪ 𝑎 ∈ 𝑏 𝑎
34	25, 31, 33	3eqtr4i 2776	. . . 4 ⊢ ((𝑈‘𝑏)‘suc ∅) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅)
35	26	itunisuc 10175	. . . . . 6 ⊢ ((𝑈‘𝑏)‘suc suc 𝑐) = ∪ ((𝑈‘𝑏)‘suc 𝑐)
36		unieq 4850	. . . . . . 7 ⊢ (((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) → ∪ ((𝑈‘𝑏)‘suc 𝑐) = ∪ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐))
37	26	itunisuc 10175	. . . . . . . . . 10 ⊢ ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐)
38	37	a1i 11	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝑏 → ((𝑈‘𝑎)‘suc 𝑐) = ∪ ((𝑈‘𝑎)‘𝑐))
39	38	iuneq2i 4945	. . . . . . . 8 ⊢ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ∪ ((𝑈‘𝑎)‘𝑐)
40		iuncom4 4932	. . . . . . . 8 ⊢ ∪ 𝑎 ∈ 𝑏 ∪ ((𝑈‘𝑎)‘𝑐) = ∪ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐)
41	39, 40	eqtr2i 2767	. . . . . . 7 ⊢ ∪ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐)
42	36, 41	eqtrdi 2794	. . . . . 6 ⊢ (((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) → ∪ ((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐))
43	35, 42	eqtrid 2790	. . . . 5 ⊢ (((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) → ((𝑈‘𝑏)‘suc suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐))
44	43	a1i 11	. . . 4 ⊢ (𝑐 ∈ ω → (((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) → ((𝑈‘𝑏)‘suc suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐)))
45	9, 14, 19, 24, 34, 44	finds 7745	. . 3 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵))
46		iun0 4991	. . . . 5 ⊢ ∪ 𝑎 ∈ 𝑏 ∅ = ∅
47	46	eqcomi 2747	. . . 4 ⊢ ∅ = ∪ 𝑎 ∈ 𝑏 ∅
48		peano2b 7729	. . . . . 6 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
49		vex 3436	. . . . . . . 8 ⊢ 𝑏 ∈ V
50	26	itunifn 10173	. . . . . . . 8 ⊢ (𝑏 ∈ V → (𝑈‘𝑏) Fn ω)
51		fndm 6536	. . . . . . . 8 ⊢ ((𝑈‘𝑏) Fn ω → dom (𝑈‘𝑏) = ω)
52	49, 50, 51	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑏) = ω
53	52	eleq2i 2830	. . . . . 6 ⊢ (suc 𝐵 ∈ dom (𝑈‘𝑏) ↔ suc 𝐵 ∈ ω)
54	48, 53	bitr4i 277	. . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈‘𝑏))
55		ndmfv 6804	. . . . 5 ⊢ (¬ suc 𝐵 ∈ dom (𝑈‘𝑏) → ((𝑈‘𝑏)‘suc 𝐵) = ∅)
56	54, 55	sylnbi 330	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑏)‘suc 𝐵) = ∅)
57		vex 3436	. . . . . . . 8 ⊢ 𝑎 ∈ V
58	26	itunifn 10173	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
59		fndm 6536	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
60	57, 58, 59	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
61	60	eleq2i 2830	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
62		ndmfv 6804	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
63	61, 62	sylnbir 331	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
64	63	iuneq2d 4953	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵) = ∪ 𝑎 ∈ 𝑏 ∅)
65	47, 56, 64	3eqtr4a 2804	. . 3 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵))
66	45, 65	pm2.61i 182	. 2 ⊢ ((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵)
67	4, 66	vtoclg 3505	1 ⊢ (𝐴 ∈ 𝑉 → ((𝑈‘𝐴)‘suc 𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  ∪ cuni 4839  ∪ ciun 4924   ↦ cmpt 5157  dom cdm 5589   ↾ cres 5591  suc csuc 6268   Fn wfn 6428  ‘cfv 6433  ωcom 7712  reccrdg 8240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241

This theorem is referenced by:  hsmexlem4  10185