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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
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