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Theorem unblem3 8099
Description: Lemma for unbnn 8101. 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 8098 . . . . . 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 6961 . . . . . . . 8 ω ⊆ On
5 sstr 3576 . . . . . . . 8 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
64, 5mpan2 703 . . . . . . 7 (𝐴 ⊆ ω → 𝐴 ⊆ On)
7 ssel 3562 . . . . . . . 8 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
87anc2li 578 . . . . . . 7 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
96, 8syl 17 . . . . . 6 (𝐴 ⊆ ω → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
109ad2antrr 758 . . . . 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 5732 . . . 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 8097 . . . . . . 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 46 . . . . 5 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴))
17 suceq 5707 . . . . . . . . 9 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
1817difeq2d 3690 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
1918inteqd 4415 . . . . . . 7 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
20 suceq 5707 . . . . . . . . 9 (𝑦 = (𝐹𝑧) → suc 𝑦 = suc (𝐹𝑧))
2120difeq2d 3690 . . . . . . . 8 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
2221inteqd 4415 . . . . . . 7 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
231, 19, 22frsucmpt2 7422 . . . . . 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 2691 . 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 1475  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cdif 3537  wss 3540   cint 4410  cmpt 4643  cres 5040  Oncon0 5640  suc csuc 5642  cfv 5804  ωcom 6957  reccrdg 7392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393
This theorem is referenced by:  unblem4  8100
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