Theorem unblem2 9240

Description: Lemma for unbnn 9243. The value of the function 𝐹 belongs to the unbounded set of natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.)

Hypothesis

Ref	Expression
unblem.2	⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)

Assertion

Ref	Expression
unblem2	⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))

Distinct variable groups:   𝑤,𝑣,𝑥,𝑧,𝐴   𝑣,𝐹,𝑤,𝑧

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unblem2

Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6842	. . . 4 ⊢ (𝑧 = ∅ → (𝐹‘𝑧) = (𝐹‘∅))
2	1	eleq1d 2822	. . 3 ⊢ (𝑧 = ∅ → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3		fveq2 6842	. . . 4 ⊢ (𝑧 = 𝑢 → (𝐹‘𝑧) = (𝐹‘𝑢))
4	3	eleq1d 2822	. . 3 ⊢ (𝑧 = 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘𝑢) ∈ 𝐴))
5		fveq2 6842	. . . 4 ⊢ (𝑧 = suc 𝑢 → (𝐹‘𝑧) = (𝐹‘suc 𝑢))
6	5	eleq1d 2822	. . 3 ⊢ (𝑧 = suc 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7		omsson 7806	. . . . . 6 ⊢ ω ⊆ On
8		sstr 3952	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
9	7, 8	mpan2 689	. . . . 5 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
10		peano1 7825	. . . . . . . . 9 ⊢ ∅ ∈ ω
11		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑤 ∈ 𝑣 ↔ ∅ ∈ 𝑣))
12	11	rexbidv 3175	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ↔ ∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣))
13	12	rspcv 3577	. . . . . . . . 9 ⊢ (∅ ∈ ω → (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣))
14	10, 13	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣)
15		df-rex 3074	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
16	14, 15	sylib 217	. . . . . . 7 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
17		exsimpl 1871	. . . . . . 7 ⊢ (∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣) → ∃𝑣 𝑣 ∈ 𝐴)
18	16, 17	syl 17	. . . . . 6 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣 𝑣 ∈ 𝐴)
19		n0 4306	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
20	18, 19	sylibr 233	. . . . 5 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → 𝐴 ≠ ∅)
21		onint 7725	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)
22	9, 20, 21	syl2an 596	. . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∩ 𝐴 ∈ 𝐴)
23		unblem.2	. . . . . . . 8 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)
24	23	fveq1i 6843	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅)
25		fr0g 8382	. . . . . . 7 ⊢ (∩ 𝐴 ∈ 𝐴 → ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅) = ∩ 𝐴)
26	24, 25	eqtr2id 2789	. . . . . 6 ⊢ (∩ 𝐴 ∈ 𝐴 → ∩ 𝐴 = (𝐹‘∅))
27	26	eleq1d 2822	. . . . 5 ⊢ (∩ 𝐴 ∈ 𝐴 → (∩ 𝐴 ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
28	27	ibi 266	. . . 4 ⊢ (∩ 𝐴 ∈ 𝐴 → (𝐹‘∅) ∈ 𝐴)
29	22, 28	syl 17	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝐹‘∅) ∈ 𝐴)
30		unblem1 9239	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴)
31		suceq 6383	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
32	31	difeq2d 4082	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
33	32	inteqd 4912	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
34		suceq 6383	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑢) → suc 𝑦 = suc (𝐹‘𝑢))
35	34	difeq2d 4082	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑢)))
36	35	inteqd 4912	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑢) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
37	23, 33, 36	frsucmpt2 8386	. . . . . . . . 9 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
38	37	eqcomd 2742	. . . . . . . 8 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) = (𝐹‘suc 𝑢))
39	38	eleq1d 2822	. . . . . . 7 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
40	39	ex 413	. . . . . 6 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
41	40	ibd 268	. . . . 5 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴))
42	30, 41	syl5 34	. . . 4 ⊢ (𝑢 ∈ ω → (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → (𝐹‘suc 𝑢) ∈ 𝐴))
43	42	expd 416	. . 3 ⊢ (𝑢 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ((𝐹‘𝑢) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴)))
44	2, 4, 6, 29, 43	finds2 7837	. 2 ⊢ (𝑧 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝐹‘𝑧) ∈ 𝐴))
45	44	com12 32	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  ∅c0 4282  ∩ cint 4907   ↦ cmpt 5188   ↾ cres 5635  Oncon0 6317  suc csuc 6319  ‘cfv 6496  ωcom 7802  reccrdg 8355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356

This theorem is referenced by:  unblem3  9241  unblem4  9242