Theorem unblem2 9251

Description: Lemma for unbnn 9254. 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 6881	. . . 4 ⊢ (𝑧 = ∅ → (𝐹‘𝑧) = (𝐹‘∅))
2	1	eleq1d 2847	. . 3 ⊢ (𝑧 = ∅ → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3		fveq2 6881	. . . 4 ⊢ (𝑧 = 𝑢 → (𝐹‘𝑧) = (𝐹‘𝑢))
4	3	eleq1d 2847	. . 3 ⊢ (𝑧 = 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘𝑢) ∈ 𝐴))
5		fveq2 6881	. . . 4 ⊢ (𝑧 = suc 𝑢 → (𝐹‘𝑧) = (𝐹‘suc 𝑢))
6	5	eleq1d 2847	. . 3 ⊢ (𝑧 = suc 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7		omsson 7864	. . . . . 6 ⊢ ω ⊆ On
8		sstr 3944	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
9	7, 8	mpan2 703	. . . . 5 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
10		peano1 7883	. . . . . . . . 9 ⊢ ∅ ∈ ω
11		eleq1 2850	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑤 ∈ 𝑣 ↔ ∅ ∈ 𝑣))
12	11	rexbidv 3188	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ↔ ∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣))
13	12	rspcv 3576	. . . . . . . . 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 3089	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
16	14, 15	sylib 221	. . . . . . 7 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
17		exsimpl 1897	. . . . . . 7 ⊢ (∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣) → ∃𝑣 𝑣 ∈ 𝐴)
18	16, 17	syl 18	. . . . . 6 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣 𝑣 ∈ 𝐴)
19		n0 4306	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
20	18, 19	sylibr 237	. . . . 5 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → 𝐴 ≠ ∅)
21		onint 7787	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)
22	9, 20, 21	syl2an 607	. . . 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 6882	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅)
25		fr0g 8421	. . . . . . 7 ⊢ (∩ 𝐴 ∈ 𝐴 → ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅) = ∩ 𝐴)
26	24, 25	eqtr2id 2810	. . . . . 6 ⊢ (∩ 𝐴 ∈ 𝐴 → ∩ 𝐴 = (𝐹‘∅))
27	26	eleq1d 2847	. . . . 5 ⊢ (∩ 𝐴 ∈ 𝐴 → (∩ 𝐴 ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
28	27	ibi 270	. . . 4 ⊢ (∩ 𝐴 ∈ 𝐴 → (𝐹‘∅) ∈ 𝐴)
29	22, 28	syl 18	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝐹‘∅) ∈ 𝐴)
30		unblem1 9250	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴)
31		suceq 6429	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
32	31	difeq2d 4080	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
33	32	inteqd 4916	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
34		suceq 6429	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑢) → suc 𝑦 = suc (𝐹‘𝑢))
35	34	difeq2d 4080	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑢)))
36	35	inteqd 4916	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑢) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
37	23, 33, 36	frsucmpt2 8425	. . . . . . . . 9 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
38	37	eqcomd 2768	. . . . . . . 8 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) = (𝐹‘suc 𝑢))
39	38	eleq1d 2847	. . . . . . 7 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
40	39	ex 417	. . . . . 6 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
41	40	ibd 272	. . . . 5 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴))
42	30, 41	syl5 35	. . . 4 ⊢ (𝑢 ∈ ω → (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → (𝐹‘suc 𝑢) ∈ 𝐴))
43	42	expd 420	. . 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 7893	. 2 ⊢ (𝑧 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝐹‘𝑧) ∈ 𝐴))
45	44	com12 33	1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3454   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  ∩ cint 4911   ↦ cmpt 5191   ↾ cres 5662  Oncon0 6360  suc csuc 6362  ‘cfv 6536  ωcom 7860  reccrdg 8394

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395

This theorem is used by:  unblem3  9252  unblem4  9253