Theorem itunisuc 10332

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 8366	. . . . . 6 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2		fvex 6840	. . . . . . 7 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3		unieq 4849	. . . . . . . 8 ⊢ (𝑎 = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) → ∪ 𝑎 = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
4		unieq 4849	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ∪ 𝑦 = ∪ 𝑎)
5	4	cbvmptv 5176	. . . . . . . 8 ⊢ (𝑦 ∈ V ↦ ∪ 𝑦) = (𝑎 ∈ V ↦ ∪ 𝑎)
6	2	uniex 7684	. . . . . . . 8 ⊢ ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
7	3, 5, 6	fvmpt 6935	. . . . . . 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 2790	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
10	9	adantl 482	. . . 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 10329	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
13	12	fveq1d 6829	. . . . 5 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
14	13	adantr 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
15	12	fveq1d 6829	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
16	15	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
17	16	unieqd 4851	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
18	10, 14, 17	3eqtr4d 2784	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
19		uni0 4866	. . . . 5 ⊢ ∪ ∅ = ∅
20	19	eqcomi 2748	. . . 4 ⊢ ∅ = ∪ ∅
21	11	itunifn 10330	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) Fn ω)
22	21	fndmd 6590	. . . . . . . . 9 ⊢ (𝐴 ∈ V → dom (𝑈‘𝐴) = ω)
23	22	eleq2d 2825	. . . . . . . 8 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ suc 𝐵 ∈ ω))
24		peano2b 7823	. . . . . . . 8 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
25	23, 24	bitr4di 290	. . . . . . 7 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
26	25	notbid 319	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
27	26	biimpar 478	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈‘𝐴))
28		ndmfv 6859	. . . . 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 2825	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
31	30	notbid 319	. . . . . . 7 ⊢ (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
32	31	biimpar 478	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈‘𝐴))
33		ndmfv 6859	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘𝐵) = ∅)
34	32, 33	syl 17	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ∅)
35	34	unieqd 4851	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ∅)
36	20, 29, 35	3eqtr4a 2800	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
37	18, 36	pm2.61dan 818	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
38		0fv 6868	. . . . 5 ⊢ (∅‘𝐵) = ∅
39	38	unieqi 4850	. . . 4 ⊢ ∪ (∅‘𝐵) = ∪ ∅
40		0fv 6868	. . . 4 ⊢ (∅‘suc 𝐵) = ∅
41	19, 39, 40	3eqtr4ri 2773	. . 3 ⊢ (∅‘suc 𝐵) = ∪ (∅‘𝐵)
42		fvprc 6819	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
43	42	fveq1d 6829	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
44	42	fveq1d 6829	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = (∅‘𝐵))
45	44	unieqd 4851	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ (∅‘𝐵))
46	41, 43, 45	3eqtr4a 2800	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
47	37, 46	pm2.61i 183	1 ⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ∅c0 4261  ∪ cuni 4838   ↦ cmpt 5153  dom cdm 5618   ↾ cres 5620  suc csuc 6312  ‘cfv 6485  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  itunitc1  10333  itunitc  10334  ituniiun  10335