MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unblem3 Structured version   Visualization version   GIF version

Theorem unblem3 8255
Description: Lemma for unbnn 8257. The value of the function 𝐹 is less than its value at a successor. (Contributed by NM, 3-Dec-2003.)
Hypothesis
Ref Expression
unblem.2 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
Assertion
Ref Expression
unblem3 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
Distinct variable groups:   𝑤,𝑣,𝑥,𝑧,𝐴   𝑣,𝐹,𝑤,𝑧
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem unblem3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 unblem.2 . . . . . . 7 𝐹 = (rec((𝑥 ∈ V ↦ (𝐴 ∖ suc 𝑥)), 𝐴) ↾ ω)
21unblem2 8254 . . . . . 6 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
32imp 444 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
4 omsson 7111 . . . . . . . 8 ω ⊆ On
5 sstr 3644 . . . . . . . 8 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
64, 5mpan2 707 . . . . . . 7 (𝐴 ⊆ ω → 𝐴 ⊆ On)
7 ssel 3630 . . . . . . . 8 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
87anc2li 579 . . . . . . 7 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
96, 8syl 17 . . . . . 6 (𝐴 ⊆ ω → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
109ad2antrr 762 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
113, 10mpd 15 . . . 4 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On))
12 onmindif 5853 . . . 4 ((𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On) → (𝐹𝑧) ∈ (𝐴 ∖ suc (𝐹𝑧)))
1311, 12syl 17 . . 3 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ (𝐴 ∖ suc (𝐹𝑧)))
14 unblem1 8253 . . . . . . 7 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑧) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴)
1514ex 449 . . . . . 6 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → ((𝐹𝑧) ∈ 𝐴 (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴))
162, 15syld 47 . . . . 5 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴))
17 suceq 5828 . . . . . . . . 9 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
1817difeq2d 3761 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
1918inteqd 4512 . . . . . . 7 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
20 suceq 5828 . . . . . . . . 9 (𝑦 = (𝐹𝑧) → suc 𝑦 = suc (𝐹𝑧))
2120difeq2d 3761 . . . . . . . 8 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
2221inteqd 4512 . . . . . . 7 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
231, 19, 22frsucmpt2 7580 . . . . . 6 ((𝑧 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧)))
2423ex 449 . . . . 5 (𝑧 ∈ ω → ( (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴 → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧))))
2516, 24sylcom 30 . . . 4 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧))))
2625imp 444 . . 3 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧)))
2713, 26eleqtrrd 2733 . 2 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
2827ex 449 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607   cint 4507  cmpt 4762  cres 5145  Oncon0 5761  suc csuc 5763  cfv 5926  ωcom 7107  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  unblem4  8256
  Copyright terms: Public domain W3C validator