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Theorem unblem2 8367
Description: Lemma for unbnn 8370. 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 6330 . . . 4 (𝑧 = ∅ → (𝐹𝑧) = (𝐹‘∅))
21eleq1d 2835 . . 3 (𝑧 = ∅ → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3 fveq2 6330 . . . 4 (𝑧 = 𝑢 → (𝐹𝑧) = (𝐹𝑢))
43eleq1d 2835 . . 3 (𝑧 = 𝑢 → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹𝑢) ∈ 𝐴))
5 fveq2 6330 . . . 4 (𝑧 = suc 𝑢 → (𝐹𝑧) = (𝐹‘suc 𝑢))
65eleq1d 2835 . . 3 (𝑧 = suc 𝑢 → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7 omsson 7214 . . . . . 6 ω ⊆ On
8 sstr 3760 . . . . . 6 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
97, 8mpan2 671 . . . . 5 (𝐴 ⊆ ω → 𝐴 ⊆ On)
10 peano1 7230 . . . . . . . . 9 ∅ ∈ ω
11 eleq1 2838 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑤𝑣 ↔ ∅ ∈ 𝑣))
1211rexbidv 3200 . . . . . . . . . 10 (𝑤 = ∅ → (∃𝑣𝐴 𝑤𝑣 ↔ ∃𝑣𝐴 ∅ ∈ 𝑣))
1312rspcv 3456 . . . . . . . . 9 (∅ ∈ ω → (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣𝐴 ∅ ∈ 𝑣))
1410, 13ax-mp 5 . . . . . . . 8 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣𝐴 ∅ ∈ 𝑣)
15 df-rex 3067 . . . . . . . 8 (∃𝑣𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣))
1614, 15sylib 208 . . . . . . 7 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣))
17 exsimpl 1946 . . . . . . 7 (∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣) → ∃𝑣 𝑣𝐴)
1816, 17syl 17 . . . . . 6 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣 𝑣𝐴)
19 n0 4078 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣𝐴)
2018, 19sylibr 224 . . . . 5 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣𝐴 ≠ ∅)
21 onint 7140 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
229, 20, 21syl2an 583 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴𝐴)
23 unblem.2 . . . . . . . 8 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
2423fveq1i 6331 . . . . . . 7 (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)‘∅)
25 fr0g 7682 . . . . . . 7 ( 𝐴𝐴 → ((rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)‘∅) = 𝐴)
2624, 25syl5req 2818 . . . . . 6 ( 𝐴𝐴 𝐴 = (𝐹‘∅))
2726eleq1d 2835 . . . . 5 ( 𝐴𝐴 → ( 𝐴𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
2827ibi 256 . . . 4 ( 𝐴𝐴 → (𝐹‘∅) ∈ 𝐴)
2922, 28syl 17 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝐹‘∅) ∈ 𝐴)
30 unblem1 8366 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑢) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴)
31 suceq 5931 . . . . . . . . . . . 12 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
3231difeq2d 3879 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
3332inteqd 4616 . . . . . . . . . 10 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
34 suceq 5931 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑢) → suc 𝑦 = suc (𝐹𝑢))
3534difeq2d 3879 . . . . . . . . . . 11 (𝑦 = (𝐹𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑢)))
3635inteqd 4616 . . . . . . . . . 10 (𝑦 = (𝐹𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑢)))
3723, 33, 36frsucmpt2 7686 . . . . . . . . 9 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = (𝐴 ∖ suc (𝐹𝑢)))
3837eqcomd 2777 . . . . . . . 8 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑢)) = (𝐹‘suc 𝑢))
3938eleq1d 2835 . . . . . . 7 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
4039ex 397 . . . . . 6 (𝑢 ∈ ω → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
4140ibd 258 . . . . 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 400 . . 3 (𝑢 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ((𝐹𝑢) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴)))
442, 4, 6, 29, 43finds2 7239 . 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 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  wss 3723  c0 4063   cint 4611  cmpt 4863  cres 5251  Oncon0 5864  suc csuc 5866  cfv 6029  ωcom 7210  reccrdg 7656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5821  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7211  df-wrecs 7557  df-recs 7619  df-rdg 7657
This theorem is referenced by:  unblem3  8368  unblem4  8369
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