Theorem unblem2 8098

Description: Lemma for unbnn 8101. 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 6103	. . . 4 ⊢ (𝑧 = ∅ → (𝐹‘𝑧) = (𝐹‘∅))
2	1	eleq1d 2672	. . 3 ⊢ (𝑧 = ∅ → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3		fveq2 6103	. . . 4 ⊢ (𝑧 = 𝑢 → (𝐹‘𝑧) = (𝐹‘𝑢))
4	3	eleq1d 2672	. . 3 ⊢ (𝑧 = 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘𝑢) ∈ 𝐴))
5		fveq2 6103	. . . 4 ⊢ (𝑧 = suc 𝑢 → (𝐹‘𝑧) = (𝐹‘suc 𝑢))
6	5	eleq1d 2672	. . 3 ⊢ (𝑧 = suc 𝑢 → ((𝐹‘𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7		omsson 6961	. . . . . 6 ⊢ ω ⊆ On
8		sstr 3576	. . . . . 6 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
9	7, 8	mpan2 703	. . . . 5 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On)
10		peano1 6977	. . . . . . . . 9 ⊢ ∅ ∈ ω
11		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (𝑤 ∈ 𝑣 ↔ ∅ ∈ 𝑣))
12	11	rexbidv 3034	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ↔ ∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣))
13	12	rspcv 3278	. . . . . . . . 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 2902	. . . . . . . 8 ⊢ (∃𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
16	14, 15	sylib 207	. . . . . . 7 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃𝑣(𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣))
17		exsimpl 1783	. . . . . . 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 3890	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
20	18, 19	sylibr 223	. . . . 5 ⊢ (∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → 𝐴 ≠ ∅)
21		onint 6887	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝐴)
22	9, 20, 21	syl2an 493	. . . 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 6104	. . . . . . 7 ⊢ (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅)
25		fr0g 7418	. . . . . . 7 ⊢ (∩ 𝐴 ∈ 𝐴 → ((rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω)‘∅) = ∩ 𝐴)
26	24, 25	syl5req 2657	. . . . . 6 ⊢ (∩ 𝐴 ∈ 𝐴 → ∩ 𝐴 = (𝐹‘∅))
27	26	eleq1d 2672	. . . . 5 ⊢ (∩ 𝐴 ∈ 𝐴 → (∩ 𝐴 ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
28	27	ibi 255	. . . 4 ⊢ (∩ 𝐴 ∈ 𝐴 → (𝐹‘∅) ∈ 𝐴)
29	22, 28	syl 17	. . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝐹‘∅) ∈ 𝐴)
30		unblem1 8097	. . . . 5 ⊢ (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴)
31		suceq 5707	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
32	31	difeq2d 3690	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
33	32	inteqd 4415	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc 𝑥))
34		suceq 5707	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑢) → suc 𝑦 = suc (𝐹‘𝑢))
35	34	difeq2d 3690	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹‘𝑢)))
36	35	inteqd 4415	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑢) → ∩ (𝐴 ∖ suc 𝑦) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
37	23, 33, 36	frsucmpt2 7422	. . . . . . . . 9 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = ∩ (𝐴 ∖ suc (𝐹‘𝑢)))
38	37	eqcomd 2616	. . . . . . . 8 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → ∩ (𝐴 ∖ suc (𝐹‘𝑢)) = (𝐹‘suc 𝑢))
39	38	eleq1d 2672	. . . . . . 7 ⊢ ((𝑢 ∈ ω ∧ ∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴) → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
40	39	ex 449	. . . . . 6 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
41	40	ibd 257	. . . . 5 ⊢ (𝑢 ∈ ω → (∩ (𝐴 ∖ suc (𝐹‘𝑢)) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴))
42	30, 41	syl5 33	. . . 4 ⊢ (𝑢 ∈ ω → (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) ∧ (𝐹‘𝑢) ∈ 𝐴) → (𝐹‘suc 𝑢) ∈ 𝐴))
43	42	expd 451	. . 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 6986	. 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   ↦ cmpt 4643   ↾ cres 5040  Oncon0 5640  suc csuc 5642  ‘cfv 5804  ωcom 6957  reccrdg 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393

This theorem is referenced by:  unblem3  8099  unblem4  8100