Theorem unblem2 9301

Description: Lemma for unbnn 9304. 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 6876	. . . 4 ⊢ (𝑧 = ∅ → (𝐹‘𝑧) = (𝐹‘∅))
2	1	eleq1d 2819	. . 3 ⊢ (𝑧 = ∅ → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3		fveq2 6876	. . . 4 ⊢ (𝑧 = 𝑢 → (𝐹‘𝑧) = (𝐹‘𝑢))
4	3	eleq1d 2819	. . 3 ⊢ (𝑧 = 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘𝑢) ∈ 𝐴))
5		fveq2 6876	. . . 4 ⊢ (𝑧 = suc 𝑢 → (𝐹‘𝑧) = (𝐹‘suc 𝑢))
6	5	eleq1d 2819	. . 3 ⊢ (𝑧 = suc 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7		omsson 7865	. . . . . 6 ⊢ ω ⊆ On
8		sstr 3967	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
9	7, 8	mpan2 691	. . . . 5 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
10		peano1 7884	. . . . . . . . 9 ⊢ ∅ ∈ ω
11		eleq1 2822	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑤 ∈ 𝑣 ↔ ∅ ∈ 𝑣))
12	11	rexbidv 3164	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ↔ ∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣))
13	12	rspcv 3597	. . . . . . . . 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 3061	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
16	14, 15	sylib 218	. . . . . . 7 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
17		exsimpl 1868	. . . . . . 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 4328	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
20	18, 19	sylibr 234	. . . . 5 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → 𝐴 ≠ ∅)
21		onint 7784	. . . . 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 6877	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅)
25		fr0g 8450	. . . . . . 7 ⊢ (∩ 𝐴 ∈ 𝐴 → ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅) = ∩ 𝐴)
26	24, 25	eqtr2id 2783	. . . . . 6 ⊢ (∩ 𝐴 ∈ 𝐴 → ∩ 𝐴 = (𝐹‘∅))
27	26	eleq1d 2819	. . . . 5 ⊢ (∩ 𝐴 ∈ 𝐴 → (∩ 𝐴 ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
28	27	ibi 267	. . . 4 ⊢ (∩ 𝐴 ∈ 𝐴 → (𝐹‘∅) ∈ 𝐴)
29	22, 28	syl 17	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝐹‘∅) ∈ 𝐴)
30		unblem1 9300	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴)
31		suceq 6419	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
32	31	difeq2d 4101	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
33	32	inteqd 4927	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
34		suceq 6419	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑢) → suc 𝑦 = suc (𝐹‘𝑢))
35	34	difeq2d 4101	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑢)))
36	35	inteqd 4927	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑢) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
37	23, 33, 36	frsucmpt2 8454	. . . . . . . . 9 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
38	37	eqcomd 2741	. . . . . . . 8 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) = (𝐹‘suc 𝑢))
39	38	eleq1d 2819	. . . . . . 7 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
40	39	ex 412	. . . . . 6 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
41	40	ibd 269	. . . . 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 415	. . 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 7894	. 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 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ∩ cint 4922   ↦ cmpt 5201   ↾ cres 5656  Oncon0 6352  suc csuc 6354  ‘cfv 6531  ωcom 7861  reccrdg 8423

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424

This theorem is referenced by:  unblem3  9302  unblem4  9303