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Theorem fin23lem41 10392
Description: Lemma for fin23 10429. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
fin23lem40.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
fin23lem41 (𝐴𝐹𝐴 ∈ FinIII)
Distinct variable groups:   𝑔,𝑎,𝑥,𝐴   𝐹,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑔)

Proof of Theorem fin23lem41
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdomi 8999 . . . . 5 (ω ≼ 𝒫 𝐴 → ∃𝑏 𝑏:ω–1-1→𝒫 𝐴)
2 fin23lem40.f . . . . . . . . . 10 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
32fin23lem33 10385 . . . . . . . . 9 (𝐴𝐹 → ∃𝑐𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)))
43adantl 481 . . . . . . . 8 ((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) → ∃𝑐𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)))
5 ssv 4008 . . . . . . . . . . 11 𝒫 𝐴 ⊆ V
6 f1ss 6809 . . . . . . . . . . 11 ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
75, 6mpan2 691 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω–1-1→V)
87ad2antrr 726 . . . . . . . . 9 (((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑))) → 𝑏:ω–1-1→V)
9 f1f 6804 . . . . . . . . . . . 12 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω⟶𝒫 𝐴)
10 frn 6743 . . . . . . . . . . . 12 (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11 uniss 4915 . . . . . . . . . . . 12 (ran 𝑏 ⊆ 𝒫 𝐴 ran 𝑏 𝒫 𝐴)
129, 10, 113syl 18 . . . . . . . . . . 11 (𝑏:ω–1-1→𝒫 𝐴 ran 𝑏 𝒫 𝐴)
13 unipw 5455 . . . . . . . . . . 11 𝒫 𝐴 = 𝐴
1412, 13sseqtrdi 4024 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴 ran 𝑏𝐴)
1514ad2antrr 726 . . . . . . . . 9 (((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑))) → ran 𝑏𝐴)
16 f1eq1 6799 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17 rneq 5947 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
1817unieqd 4920 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran 𝑑 = ran 𝑒)
1918sseq1d 4015 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran 𝑑𝐴 ran 𝑒𝐴))
2016, 19anbi12d 632 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) ↔ (𝑒:ω–1-1→V ∧ ran 𝑒𝐴)))
21 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (𝑐𝑑) = (𝑐𝑒))
22 f1eq1 6799 . . . . . . . . . . . . . . 15 ((𝑐𝑑) = (𝑐𝑒) → ((𝑐𝑑):ω–1-1→V ↔ (𝑐𝑒):ω–1-1→V))
2321, 22syl 17 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ((𝑐𝑑):ω–1-1→V ↔ (𝑐𝑒):ω–1-1→V))
2421rneqd 5949 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran (𝑐𝑑) = ran (𝑐𝑒))
2524unieqd 4920 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran (𝑐𝑑) = ran (𝑐𝑒))
2625, 18psseq12d 4097 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran (𝑐𝑑) ⊊ ran 𝑑 ran (𝑐𝑒) ⊊ ran 𝑒))
2723, 26anbi12d 632 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → (((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑) ↔ ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
2820, 27imbi12d 344 . . . . . . . . . . . 12 (𝑑 = 𝑒 → (((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)) ↔ ((𝑒:ω–1-1→V ∧ ran 𝑒𝐴) → ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒))))
2928cbvalvw 2035 . . . . . . . . . . 11 (∀𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)) ↔ ∀𝑒((𝑒:ω–1-1→V ∧ ran 𝑒𝐴) → ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
3029biimpi 216 . . . . . . . . . 10 (∀𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)) → ∀𝑒((𝑒:ω–1-1→V ∧ ran 𝑒𝐴) → ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
3130adantl 481 . . . . . . . . 9 (((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑))) → ∀𝑒((𝑒:ω–1-1→V ∧ ran 𝑒𝐴) → ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
32 eqid 2737 . . . . . . . . 9 (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
332, 8, 15, 31, 32fin23lem39 10390 . . . . . . . 8 (((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑))) → ¬ 𝐴𝐹)
344, 33exlimddv 1935 . . . . . . 7 ((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) → ¬ 𝐴𝐹)
3534pm2.01da 799 . . . . . 6 (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
3635exlimiv 1930 . . . . 5 (∃𝑏 𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
371, 36syl 17 . . . 4 (ω ≼ 𝒫 𝐴 → ¬ 𝐴𝐹)
3837con2i 139 . . 3 (𝐴𝐹 → ¬ ω ≼ 𝒫 𝐴)
39 pwexg 5378 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
40 isfin4-2 10354 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4139, 40syl 17 . . 3 (𝐴𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4238, 41mpbird 257 . 2 (𝐴𝐹 → 𝒫 𝐴 ∈ FinIV)
43 isfin3 10336 . 2 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
4442, 43sylibr 234 1 (𝐴𝐹𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  wss 3951  wpss 3952  𝒫 cpw 4600   cuni 4907   cint 4946   class class class wbr 5143  ran crn 5686  cres 5687  suc csuc 6386  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  ωcom 7887  reccrdg 8449  m cmap 8866  cdom 8983  FinIVcfin4 10320  FinIIIcfin3 10321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-fin4 10327  df-fin3 10328
This theorem is referenced by:  isf33lem  10406
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