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Theorem unblem3 9296
Description: Lemma for unbnn 9298. 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 9295 . . . . . 6 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
32imp 406 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
4 omsson 7855 . . . . . . . 8 ω ⊆ On
5 sstr 3985 . . . . . . . 8 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
64, 5mpan2 688 . . . . . . 7 (𝐴 ⊆ ω → 𝐴 ⊆ On)
7 ssel 3970 . . . . . . . 8 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
87anc2li 555 . . . . . . 7 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
96, 8syl 17 . . . . . 6 (𝐴 ⊆ ω → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
109ad2antrr 723 . . . . 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 6449 . . . 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 9294 . . . . . . 7 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑧) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴)
1514ex 412 . . . . . 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 6423 . . . . . . . . 9 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
1817difeq2d 4117 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
1918inteqd 4948 . . . . . . 7 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
20 suceq 6423 . . . . . . . . 9 (𝑦 = (𝐹𝑧) → suc 𝑦 = suc (𝐹𝑧))
2120difeq2d 4117 . . . . . . . 8 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
2221inteqd 4948 . . . . . . 7 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
231, 19, 22frsucmpt2 8438 . . . . . 6 ((𝑧 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧)))
2423ex 412 . . . . 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 406 . . 3 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧)))
2713, 26eleqtrrd 2830 . 2 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
2827ex 412 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  Vcvv 3468  cdif 3940  wss 3943   cint 4943  cmpt 5224  cres 5671  Oncon0 6357  suc csuc 6359  cfv 6536  ωcom 7851  reccrdg 8407
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408
This theorem is referenced by:  unblem4  9297
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