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Theorem unblem3 9248
Description: Lemma for unbnn 9250. 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 9247 . . . . . 6 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ 𝐴))
32imp 408 . . . . 5 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ 𝐴)
4 omsson 7811 . . . . . . . 8 ω ⊆ On
5 sstr 3957 . . . . . . . 8 ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On)
64, 5mpan2 690 . . . . . . 7 (𝐴 ⊆ ω → 𝐴 ⊆ On)
7 ssel 3942 . . . . . . . 8 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐹𝑧) ∈ On))
87anc2li 557 . . . . . . 7 (𝐴 ⊆ On → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
96, 8syl 17 . . . . . 6 (𝐴 ⊆ ω → ((𝐹𝑧) ∈ 𝐴 → (𝐴 ⊆ On ∧ (𝐹𝑧) ∈ On)))
109ad2antrr 725 . . . . 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 6414 . . . 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 9246 . . . . . . 7 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ (𝐹𝑧) ∈ 𝐴) → (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴)
1514ex 414 . . . . . 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 6388 . . . . . . . . 9 (𝑦 = 𝑥 → suc 𝑦 = suc 𝑥)
1817difeq2d 4087 . . . . . . . 8 (𝑦 = 𝑥 → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
1918inteqd 4917 . . . . . . 7 (𝑦 = 𝑥 (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc 𝑥))
20 suceq 6388 . . . . . . . . 9 (𝑦 = (𝐹𝑧) → suc 𝑦 = suc (𝐹𝑧))
2120difeq2d 4087 . . . . . . . 8 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
2221inteqd 4917 . . . . . . 7 (𝑦 = (𝐹𝑧) → (𝐴 ∖ suc 𝑦) = (𝐴 ∖ suc (𝐹𝑧)))
231, 19, 22frsucmpt2 8391 . . . . . 6 ((𝑧 ∈ ω ∧ (𝐴 ∖ suc (𝐹𝑧)) ∈ 𝐴) → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧)))
2423ex 414 . . . . 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 408 . . 3 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹‘suc 𝑧) = (𝐴 ∖ suc (𝐹𝑧)))
2713, 26eleqtrrd 2841 . 2 (((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) ∧ 𝑧 ∈ ω) → (𝐹𝑧) ∈ (𝐹‘suc 𝑧))
2827ex 414 1 ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣𝐴 𝑤𝑣) → (𝑧 ∈ ω → (𝐹𝑧) ∈ (𝐹‘suc 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  wss 3915   cint 4912  cmpt 5193  cres 5640  Oncon0 6322  suc csuc 6324  cfv 6501  ωcom 7807  reccrdg 8360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361
This theorem is referenced by:  unblem4  9249
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