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Theorem unblem2 9329
Description: Lemma for unbnn 9332. 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 ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 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.
StepHypRef Expression
1 fveq2 6906 . . . 4 (𝑧 = ∅ → (𝐹𝑧) = (𝐹‘∅))
21eleq1d 2826 . . 3 (𝑧 = ∅ → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3 fveq2 6906 . . . 4 (𝑧 = 𝑢 → (𝐹𝑧) = (𝐹𝑢))
43eleq1d 2826 . . 3 (𝑧 = 𝑢 → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹𝑢) ∈ 𝐴))
5 fveq2 6906 . . . 4 (𝑧 = suc 𝑢 → (𝐹𝑧) = (𝐹‘suc 𝑢))
65eleq1d 2826 . . 3 (𝑧 = suc 𝑢 → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7 omsson 7891 . . . . . 6 ω ⊆ On
8 sstr 3992 . . . . . 6 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
97, 8mpan2 691 . . . . 5 (𝐴 ⊆ ω → 𝐴 ⊆ On)
10 peano1 7910 . . . . . . . . 9 ∅ ∈ ω
11 eleq1 2829 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑤𝑣 ↔ ∅ ∈ 𝑣))
1211rexbidv 3179 . . . . . . . . . 10 (𝑤 = ∅ → (∃𝑣𝐴 𝑤𝑣 ↔ ∃𝑣𝐴 ∅ ∈ 𝑣))
1312rspcv 3618 . . . . . . . . 9 (∅ ∈ ω → (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣𝐴 ∅ ∈ 𝑣))
1410, 13ax-mp 5 . . . . . . . 8 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣𝐴 ∅ ∈ 𝑣)
15 df-rex 3071 . . . . . . . 8 (∃𝑣𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣))
1614, 15sylib 218 . . . . . . 7 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣))
17 exsimpl 1868 . . . . . . 7 (∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣) → ∃𝑣 𝑣𝐴)
1816, 17syl 17 . . . . . 6 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣 𝑣𝐴)
19 n0 4353 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣𝐴)
2018, 19sylibr 234 . . . . 5 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣𝐴 ≠ ∅)
21 onint 7810 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
229, 20, 21syl2an 596 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴𝐴)
23 unblem.2 . . . . . . . 8 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
2423fveq1i 6907 . . . . . . 7 (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)‘∅)
25 fr0g 8476 . . . . . . 7 ( 𝐴𝐴 → ((rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)‘∅) = 𝐴)
2624, 25eqtr2id 2790 . . . . . 6 ( 𝐴𝐴 𝐴 = (𝐹‘∅))
2726eleq1d 2826 . . . . 5 ( 𝐴𝐴 → ( 𝐴𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
2827ibi 267 . . . 4 ( 𝐴𝐴 → (𝐹‘∅) ∈ 𝐴)
2922, 28syl 17 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝐹‘∅) ∈ 𝐴)
30 unblem1 9328 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑢) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴)
31 suceq 6450 . . . . . . . . . . . 12 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
3231difeq2d 4126 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
3332inteqd 4951 . . . . . . . . . 10 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
34 suceq 6450 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑢) → suc 𝑦 = suc (𝐹𝑢))
3534difeq2d 4126 . . . . . . . . . . 11 (𝑦 = (𝐹𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑢)))
3635inteqd 4951 . . . . . . . . . 10 (𝑦 = (𝐹𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑢)))
3723, 33, 36frsucmpt2 8480 . . . . . . . . 9 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = (𝐴 ∖ suc (𝐹𝑢)))
3837eqcomd 2743 . . . . . . . 8 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑢)) = (𝐹‘suc 𝑢))
3938eleq1d 2826 . . . . . . 7 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
4039ex 412 . . . . . 6 (𝑢 ∈ ω → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
4140ibd 269 . . . . 5 (𝑢 ∈ ω → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴))
4230, 41syl5 34 . . . 4 (𝑢 ∈ ω → (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑢) ∈ 𝐴) → (𝐹‘suc 𝑢) ∈ 𝐴))
4342expd 415 . . 3 (𝑢 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ((𝐹𝑢) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴)))
442, 4, 6, 29, 43finds2 7920 . 2 (𝑧 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝐹𝑧) ∈ 𝐴))
4544com12 32 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  c0 4333   cint 4946  cmpt 5225  cres 5687  Oncon0 6384  suc csuc 6386  cfv 6561  ωcom 7887  reccrdg 8449
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  unblem3  9330  unblem4  9331
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