Theorem itunisuc 10348

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 8382	. . . . . 6 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2		fvex 6853	. . . . . . 7 ⊢ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
3		unieq 4878	. . . . . . . 8 ⊢ (𝑎 = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) → ∪ 𝑎 = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
4		unieq 4878	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ∪ 𝑦 = ∪ 𝑎)
5	4	cbvmptv 5206	. . . . . . . 8 ⊢ (𝑦 ∈ V ↦ ∪ 𝑦) = (𝑎 ∈ V ↦ ∪ 𝑎)
6	2	uniex 7697	. . . . . . . 8 ⊢ ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
7	3, 5, 6	fvmpt 6950	. . . . . . 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 2780	. . . . 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 10345	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
13	12	fveq1d 6842	. . . . 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 6842	. . . . . 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 4880	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
18	10, 14, 17	3eqtr4d 2774	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
19		uni0 4895	. . . . 5 ⊢ ∪ ∅ = ∅
20	19	eqcomi 2738	. . . 4 ⊢ ∅ = ∪ ∅
21	11	itunifn 10346	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) Fn ω)
22	21	fndmd 6605	. . . . . . . . 9 ⊢ (𝐴 ∈ V → dom (𝑈‘𝐴) = ω)
23	22	eleq2d 2814	. . . . . . . 8 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ suc 𝐵 ∈ ω))
24		peano2b 7839	. . . . . . . 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 6875	. . . . 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 2814	. . . . . . . 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 6875	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘𝐵) = ∅)
34	32, 33	syl 17	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘𝐵) = ∅)
35	34	unieqd 4880	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ ∅)
36	20, 29, 35	3eqtr4a 2790	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
37	18, 36	pm2.61dan 812	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
38		0fv 6884	. . . . 5 ⊢ (∅‘𝐵) = ∅
39	38	unieqi 4879	. . . 4 ⊢ ∪ (∅‘𝐵) = ∪ ∅
40		0fv 6884	. . . 4 ⊢ (∅‘suc 𝐵) = ∅
41	19, 39, 40	3eqtr4ri 2763	. . 3 ⊢ (∅‘suc 𝐵) = ∪ (∅‘𝐵)
42		fvprc 6832	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
43	42	fveq1d 6842	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
44	42	fveq1d 6842	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = (∅‘𝐵))
45	44	unieqd 4880	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ (∅‘𝐵))
46	41, 43, 45	3eqtr4a 2790	. 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 2109  Vcvv 3444  ∅c0 4292  ∪ cuni 4867   ↦ cmpt 5183  dom cdm 5631   ↾ cres 5633  suc csuc 6322  ‘cfv 6499  ωcom 7822  reccrdg 8354

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-inf2 9570

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355

This theorem is referenced by:  itunitc1  10349  itunitc  10350  ituniiun  10351