Theorem itunisuc 10459

Description: Successor iterated union. (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
itunisuc	⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)

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

Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunisuc

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frsuc 8477	. . . . . 6 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2		fvex 6919	. . . . . . 7 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3		unieq 4918	. . . . . . . 8 ⊢ (𝑎 = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) → ∪ 𝑎 = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
4		unieq 4918	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ∪ 𝑦 = ∪ 𝑎)
5	4	cbvmptv 5255	. . . . . . . 8 ⊢ (𝑦 ∈ V ↦ ∪ 𝑦) = (𝑎 ∈ V ↦ ∪ 𝑎)
6	2	uniex 7761	. . . . . . . 8 ⊢ ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
7	3, 5, 6	fvmpt 7016	. . . . . . 7 ⊢ (((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V → ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
8	2, 7	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)
9	1, 8	eqtrdi 2793	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
10	9	adantl 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
11		ituni.u	. . . . . . 7 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
12	11	itunifval 10456	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
13	12	fveq1d 6908	. . . . 5 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
14	13	adantr 480	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
15	12	fveq1d 6908	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
16	15	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
17	16	unieqd 4920	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
18	10, 14, 17	3eqtr4d 2787	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
19		uni0 4935	. . . . 5 ⊢ ∪ ∅ = ∅
20	19	eqcomi 2746	. . . 4 ⊢ ∅ = ∪ ∅
21	11	itunifn 10457	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) Fn ω)
22	21	fndmd 6673	. . . . . . . . 9 ⊢ (𝐴 ∈ V → dom (𝑈‘𝐴) = ω)
23	22	eleq2d 2827	. . . . . . . 8 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ suc 𝐵 ∈ ω))
24		peano2b 7904	. . . . . . . 8 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
25	23, 24	bitr4di 289	. . . . . . 7 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
26	25	notbid 318	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
27	26	biimpar 477	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈‘𝐴))
28		ndmfv 6941	. . . . 5 ⊢ (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘suc 𝐵) = ∅)
29	27, 28	syl 17	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∅)
30	22	eleq2d 2827	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
31	30	notbid 318	. . . . . . 7 ⊢ (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
32	31	biimpar 477	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈‘𝐴))
33		ndmfv 6941	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘𝐵) = ∅)
34	32, 33	syl 17	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ∅)
35	34	unieqd 4920	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ∅)
36	20, 29, 35	3eqtr4a 2803	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
37	18, 36	pm2.61dan 813	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
38		0fv 6950	. . . . 5 ⊢ (∅‘𝐵) = ∅
39	38	unieqi 4919	. . . 4 ⊢ ∪ (∅‘𝐵) = ∪ ∅
40		0fv 6950	. . . 4 ⊢ (∅‘suc 𝐵) = ∅
41	19, 39, 40	3eqtr4ri 2776	. . 3 ⊢ (∅‘suc 𝐵) = ∪ (∅‘𝐵)
42		fvprc 6898	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
43	42	fveq1d 6908	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
44	42	fveq1d 6908	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = (∅‘𝐵))
45	44	unieqd 4920	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ (∅‘𝐵))
46	41, 43, 45	3eqtr4a 2803	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
47	37, 46	pm2.61i 182	1 ⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ∪ cuni 4907   ↦ cmpt 5225  dom cdm 5685   ↾ cres 5687  suc csuc 6386  ‘cfv 6561  ωcom 7887  reccrdg 8449

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450

This theorem is referenced by:  itunitc1  10460  itunitc  10461  ituniiun  10462