Theorem itunisuc 10497

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 8445	. . . . . 6 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ((𝑦 ∈ V ↦ ∪ 𝑦)‘((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵)))
2		fvex 6898	. . . . . . 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 5209	. . . . . . . 8 ⊢ (𝑦 ∈ V ↦ ∪ 𝑦) = (𝑎 ∈ V ↦ ∪ 𝑎)
6	2	uniex 7758	. . . . . . . 8 ⊢ ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵) ∈ V
7	3, 5, 6	fvmpt 6993	. . . . . . 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 2812	. . . . 5 ⊢ (𝐵 ∈ ω → ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵) = ∪ ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
10	9	adantl 487	. . . 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 10494	. . . . . 6 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) = (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω))
13	12	fveq1d 6887	. . . . 5 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
14	13	adantr 486	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘suc 𝐵))
15	12	fveq1d 6887	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = ((rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝐴) ↾ ω)‘𝐵))
16	15	adantr 486	. . . . 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 2806	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
19		uni0 4896	. . . . 5 ⊢ ∪ ∅ = ∅
20	19	eqcomi 2770	. . . 4 ⊢ ∅ = ∪ ∅
21	11	itunifn 10495	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝑈‘𝐴) Fn ω)
22	21	fndmd 6644	. . . . . . . . 9 ⊢ (𝐴 ∈ V → dom (𝑈‘𝐴) = ω)
23	22	eleq2d 2847	. . . . . . . 8 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ suc 𝐵 ∈ ω))
24		peano2b 7894	. . . . . . . 8 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
25	23, 24	bitr4di 292	. . . . . . 7 ⊢ (𝐴 ∈ V → (suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
26	25	notbid 321	. . . . . 6 ⊢ (𝐴 ∈ V → (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
27	26	biimpar 483	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ suc 𝐵 ∈ dom (𝑈‘𝐴))
28		ndmfv 6917	. . . . 5 ⊢ (¬ suc 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘suc 𝐵) = ∅)
29	27, 28	syl 18	. . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∅)
30	22	eleq2d 2847	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐵 ∈ dom (𝑈‘𝐴) ↔ 𝐵 ∈ ω))
31	30	notbid 321	. . . . . . 7 ⊢ (𝐴 ∈ V → (¬ 𝐵 ∈ dom (𝑈‘𝐴) ↔ ¬ 𝐵 ∈ ω))
32	31	biimpar 483	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ¬ 𝐵 ∈ dom (𝑈‘𝐴))
33		ndmfv 6917	. . . . . 6 ⊢ (¬ 𝐵 ∈ dom (𝑈‘𝐴) → ((𝑈‘𝐴)‘𝐵) = ∅)
34	32, 33	syl 18	. . . . 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 2822	. . 3 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω) → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
37	18, 36	pm2.61dan 825	. 2 ⊢ (𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
38		0fv 6926	. . . . 5 ⊢ (∅‘𝐵) = ∅
39	38	unieqi 4879	. . . 4 ⊢ ∪ (∅‘𝐵) = ∪ ∅
40		0fv 6926	. . . 4 ⊢ (∅‘suc 𝐵) = ∅
41	19, 39, 40	3eqtr4ri 2795	. . 3 ⊢ (∅‘suc 𝐵) = ∪ (∅‘𝐵)
42		fvprc 6877	. . . 4 ⊢ (¬ 𝐴 ∈ V → (𝑈‘𝐴) = ∅)
43	42	fveq1d 6887	. . 3 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = (∅‘suc 𝐵))
44	42	fveq1d 6887	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘𝐵) = (∅‘𝐵))
45	44	unieqd 4880	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ ((𝑈‘𝐴)‘𝐵) = ∪ (∅‘𝐵))
46	41, 43, 45	3eqtr4a 2822	. 2 ⊢ (¬ 𝐴 ∈ V → ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵))
47	37, 46	pm2.61i 184	1 ⊢ ((𝑈‘𝐴)‘suc 𝐵) = ∪ ((𝑈‘𝐴)‘𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  ∪ cuni 4867   ↦ cmpt 5186  dom cdm 5651   ↾ cres 5653  suc csuc 6364  ‘cfv 6538  ωcom 7877  reccrdg 8417

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418

This theorem is used by:  itunitc1  10498  itunitc  10499  ituniiun  10500