Theorem ituniiun 10382

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 6861	. . . 4 ⊢ (𝑏 = 𝐴 → (𝑈‘𝑏) = (𝑈‘𝐴))
2	1	fveq1d 6863	. . 3 ⊢ (𝑏 = 𝐴 → ((𝑈‘𝑏)‘suc 𝐵) = ((𝑈‘𝐴)‘suc 𝐵))
3		iuneq1 4975	. . 3 ⊢ (𝑏 = 𝐴 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵))
4	2, 3	eqeq12d 2746	. 2 ⊢ (𝑏 = 𝐴 → (((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵) ↔ ((𝑈‘𝐴)‘suc 𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵)))
5		suceq 6403	. . . . . 6 ⊢ (𝑑 = ∅ → suc 𝑑 = suc ∅)
6	5	fveq2d 6865	. . . . 5 ⊢ (𝑑 = ∅ → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc ∅))
7		fveq2 6861	. . . . . 6 ⊢ (𝑑 = ∅ → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘∅))
8	7	iuneq2d 4989	. . . . 5 ⊢ (𝑑 = ∅ → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅))
9	6, 8	eqeq12d 2746	. . . 4 ⊢ (𝑑 = ∅ → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc ∅) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅)))
10		suceq 6403	. . . . . 6 ⊢ (𝑑 = 𝑐 → suc 𝑑 = suc 𝑐)
11	10	fveq2d 6865	. . . . 5 ⊢ (𝑑 = 𝑐 → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc 𝑐))
12		fveq2 6861	. . . . . 6 ⊢ (𝑑 = 𝑐 → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘𝑐))
13	12	iuneq2d 4989	. . . . 5 ⊢ (𝑑 = 𝑐 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐))
14	11, 13	eqeq12d 2746	. . . 4 ⊢ (𝑑 = 𝑐 → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐)))
15		suceq 6403	. . . . . 6 ⊢ (𝑑 = suc 𝑐 → suc 𝑑 = suc suc 𝑐)
16	15	fveq2d 6865	. . . . 5 ⊢ (𝑑 = suc 𝑐 → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc suc 𝑐))
17		fveq2 6861	. . . . . 6 ⊢ (𝑑 = suc 𝑐 → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘suc 𝑐))
18	17	iuneq2d 4989	. . . . 5 ⊢ (𝑑 = suc 𝑐 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐))
19	16, 18	eqeq12d 2746	. . . 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 6403	. . . . . 6 ⊢ (𝑑 = 𝐵 → suc 𝑑 = suc 𝐵)
21	20	fveq2d 6865	. . . . 5 ⊢ (𝑑 = 𝐵 → ((𝑈‘𝑏)‘suc 𝑑) = ((𝑈‘𝑏)‘suc 𝐵))
22		fveq2 6861	. . . . . 6 ⊢ (𝑑 = 𝐵 → ((𝑈‘𝑎)‘𝑑) = ((𝑈‘𝑎)‘𝐵))
23	22	iuneq2d 4989	. . . . 5 ⊢ (𝑑 = 𝐵 → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵))
24	21, 23	eqeq12d 2746	. . . 4 ⊢ (𝑑 = 𝐵 → (((𝑈‘𝑏)‘suc 𝑑) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑑) ↔ ((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵)))
25		uniiun 5025	. . . . 5 ⊢ ∪ 𝑏 = ∪ 𝑎 ∈ 𝑏 𝑎
26		ituni.u	. . . . . . 7 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
27	26	itunisuc 10379	. . . . . 6 ⊢ ((𝑈‘𝑏)‘suc ∅) = ∪ ((𝑈‘𝑏)‘∅)
28	26	ituni0 10378	. . . . . . . 8 ⊢ (𝑏 ∈ V → ((𝑈‘𝑏)‘∅) = 𝑏)
29	28	elv 3455	. . . . . . 7 ⊢ ((𝑈‘𝑏)‘∅) = 𝑏
30	29	unieqi 4886	. . . . . 6 ⊢ ∪ ((𝑈‘𝑏)‘∅) = ∪ 𝑏
31	27, 30	eqtri 2753	. . . . 5 ⊢ ((𝑈‘𝑏)‘suc ∅) = ∪ 𝑏
32	26	ituni0 10378	. . . . . 6 ⊢ (𝑎 ∈ 𝑏 → ((𝑈‘𝑎)‘∅) = 𝑎)
33	32	iuneq2i 4980	. . . . 5 ⊢ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅) = ∪ 𝑎 ∈ 𝑏 𝑎
34	25, 31, 33	3eqtr4i 2763	. . . 4 ⊢ ((𝑈‘𝑏)‘suc ∅) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘∅)
35	26	itunisuc 10379	. . . . . 6 ⊢ ((𝑈‘𝑏)‘suc suc 𝑐) = ∪ ((𝑈‘𝑏)‘suc 𝑐)
36		unieq 4885	. . . . . . 7 ⊢ (((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) → ∪ ((𝑈‘𝑏)‘suc 𝑐) = ∪ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐))
37	26	itunisuc 10379	. . . . . . . . . 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 4980	. . . . . . . 8 ⊢ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ∪ ((𝑈‘𝑎)‘𝑐)
40		iuncom4 4967	. . . . . . . 8 ⊢ ∪ 𝑎 ∈ 𝑏 ∪ ((𝑈‘𝑎)‘𝑐) = ∪ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐)
41	39, 40	eqtr2i 2754	. . . . . . 7 ⊢ ∪ ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐)
42	36, 41	eqtrdi 2781	. . . . . 6 ⊢ (((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝑐) → ∪ ((𝑈‘𝑏)‘suc 𝑐) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘suc 𝑐))
43	35, 42	eqtrid 2777	. . . . 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 7875	. . 3 ⊢ (𝐵 ∈ ω → ((𝑈‘𝑏)‘suc 𝐵) = ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵))
46		iun0 5029	. . . . 5 ⊢ ∪ 𝑎 ∈ 𝑏 ∅ = ∅
47	46	eqcomi 2739	. . . 4 ⊢ ∅ = ∪ 𝑎 ∈ 𝑏 ∅
48		peano2b 7862	. . . . . 6 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
49		vex 3454	. . . . . . . 8 ⊢ 𝑏 ∈ V
50	26	itunifn 10377	. . . . . . . 8 ⊢ (𝑏 ∈ V → (𝑈‘𝑏) Fn ω)
51		fndm 6624	. . . . . . . 8 ⊢ ((𝑈‘𝑏) Fn ω → dom (𝑈‘𝑏) = ω)
52	49, 50, 51	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑏) = ω
53	52	eleq2i 2821	. . . . . 6 ⊢ (suc 𝐵 ∈ dom (𝑈‘𝑏) ↔ suc 𝐵 ∈ ω)
54	48, 53	bitr4i 278	. . . . 5 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ dom (𝑈‘𝑏))
55		ndmfv 6896	. . . . 5 ⊢ (¬ suc 𝐵 ∈ dom (𝑈‘𝑏) → ((𝑈‘𝑏)‘suc 𝐵) = ∅)
56	54, 55	sylnbi 330	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑏)‘suc 𝐵) = ∅)
57		vex 3454	. . . . . . . 8 ⊢ 𝑎 ∈ V
58	26	itunifn 10377	. . . . . . . 8 ⊢ (𝑎 ∈ V → (𝑈‘𝑎) Fn ω)
59		fndm 6624	. . . . . . . 8 ⊢ ((𝑈‘𝑎) Fn ω → dom (𝑈‘𝑎) = ω)
60	57, 58, 59	mp2b 10	. . . . . . 7 ⊢ dom (𝑈‘𝑎) = ω
61	60	eleq2i 2821	. . . . . 6 ⊢ (𝐵 ∈ dom (𝑈‘𝑎) ↔ 𝐵 ∈ ω)
62		ndmfv 6896	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝑎) → ((𝑈‘𝑎)‘𝐵) = ∅)
63	61, 62	sylnbir 331	. . . . 5 ⊢ (¬ 𝐵 ∈ ω → ((𝑈‘𝑎)‘𝐵) = ∅)
64	63	iuneq2d 4989	. . . 4 ⊢ (¬ 𝐵 ∈ ω → ∪ 𝑎 ∈ 𝑏 ((𝑈‘𝑎)‘𝐵) = ∪ 𝑎 ∈ 𝑏 ∅)
65	47, 56, 64	3eqtr4a 2791	. . 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 3523	1 ⊢ (𝐴 ∈ 𝑉 → ((𝑈‘𝐴)‘suc 𝐵) = ∪ 𝑎 ∈ 𝐴 ((𝑈‘𝑎)‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  ∪ cuni 4874  ∪ ciun 4958   ↦ cmpt 5191  dom cdm 5641   ↾ cres 5643  suc csuc 6337   Fn wfn 6509  ‘cfv 6514  ωcom 7845  reccrdg 8380

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-inf2 9601

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381

This theorem is referenced by:  hsmexlem4  10389