Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unblem2 Structured version   Visualization version   GIF version

Theorem unblem2 8760
 Description: Lemma for unbnn 8763. 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 6667 . . . 4 (𝑧 = ∅ → (𝐹𝑧) = (𝐹‘∅))
21eleq1d 2902 . . 3 (𝑧 = ∅ → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
3 fveq2 6667 . . . 4 (𝑧 = 𝑢 → (𝐹𝑧) = (𝐹𝑢))
43eleq1d 2902 . . 3 (𝑧 = 𝑢 → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹𝑢) ∈ 𝐴))
5 fveq2 6667 . . . 4 (𝑧 = suc 𝑢 → (𝐹𝑧) = (𝐹‘suc 𝑢))
65eleq1d 2902 . . 3 (𝑧 = suc 𝑢 → ((𝐹𝑧) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
7 omsson 7572 . . . . . 6 ω ⊆ On
8 sstr 3979 . . . . . 6 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
97, 8mpan2 687 . . . . 5 (𝐴 ⊆ ω → 𝐴 ⊆ On)
10 peano1 7589 . . . . . . . . 9 ∅ ∈ ω
11 eleq1 2905 . . . . . . . . . . 11 (𝑤 = ∅ → (𝑤𝑣 ↔ ∅ ∈ 𝑣))
1211rexbidv 3302 . . . . . . . . . 10 (𝑤 = ∅ → (∃𝑣𝐴 𝑤𝑣 ↔ ∃𝑣𝐴 ∅ ∈ 𝑣))
1312rspcv 3622 . . . . . . . . 9 (∅ ∈ ω → (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣𝐴 ∅ ∈ 𝑣))
1410, 13ax-mp 5 . . . . . . . 8 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣𝐴 ∅ ∈ 𝑣)
15 df-rex 3149 . . . . . . . 8 (∃𝑣𝐴 ∅ ∈ 𝑣 ↔ ∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣))
1614, 15sylib 219 . . . . . . 7 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣))
17 exsimpl 1862 . . . . . . 7 (∃𝑣(𝑣𝐴 ∧ ∅ ∈ 𝑣) → ∃𝑣 𝑣𝐴)
1816, 17syl 17 . . . . . 6 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣 → ∃𝑣 𝑣𝐴)
19 n0 4314 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣𝐴)
2018, 19sylibr 235 . . . . 5 (∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣𝐴 ≠ ∅)
21 onint 7498 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
229, 20, 21syl2an 595 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → 𝐴𝐴)
23 unblem.2 . . . . . . . 8 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
2423fveq1i 6668 . . . . . . 7 (𝐹‘∅) = ((rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)‘∅)
25 fr0g 8062 . . . . . . 7 ( 𝐴𝐴 → ((rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)‘∅) = 𝐴)
2624, 25syl5req 2874 . . . . . 6 ( 𝐴𝐴 𝐴 = (𝐹‘∅))
2726eleq1d 2902 . . . . 5 ( 𝐴𝐴 → ( 𝐴𝐴 ↔ (𝐹‘∅) ∈ 𝐴))
2827ibi 268 . . . 4 ( 𝐴𝐴 → (𝐹‘∅) ∈ 𝐴)
2922, 28syl 17 . . 3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝐹‘∅) ∈ 𝐴)
30 unblem1 8759 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑢) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴)
31 suceq 6254 . . . . . . . . . . . 12 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
3231difeq2d 4103 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
3332inteqd 4879 . . . . . . . . . 10 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
34 suceq 6254 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑢) → suc 𝑦 = suc (𝐹𝑢))
3534difeq2d 4103 . . . . . . . . . . 11 (𝑦 = (𝐹𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑢)))
3635inteqd 4879 . . . . . . . . . 10 (𝑦 = (𝐹𝑢) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑢)))
3723, 33, 36frsucmpt2 8067 . . . . . . . . 9 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → (𝐹‘suc 𝑢) = (𝐴 ∖ suc (𝐹𝑢)))
3837eqcomd 2832 . . . . . . . 8 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑢)) = (𝐹‘suc 𝑢))
3938eleq1d 2902 . . . . . . 7 ((𝑢 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴) → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴))
4039ex 413 . . . . . 6 (𝑢 ∈ ω → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 → ( (𝐴 ∖ suc (𝐹𝑢)) ∈ 𝐴 ↔ (𝐹‘suc 𝑢) ∈ 𝐴)))
4140ibd 270 . . . . 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 416 . . 3 (𝑢 ∈ ω → ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ((𝐹𝑢) ∈ 𝐴 → (𝐹‘suc 𝑢) ∈ 𝐴)))
442, 4, 6, 29, 43finds2 7598 . 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 207   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ∖ cdif 3937   ⊆ wss 3940  ∅c0 4295  ∩ cint 4874   ↦ cmpt 5143   ↾ cres 5556  Oncon0 6189  suc csuc 6191  ‘cfv 6352  ωcom 7568  reccrdg 8036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037 This theorem is referenced by:  unblem3  8761  unblem4  8762
 Copyright terms: Public domain W3C validator