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Theorem fin23lem41 9057
Description: Lemma for fin23 9094. 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 7852 . . . . 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 9050 . . . . . . . . 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 3588 . . . . . . . . . . 11 𝒫 𝐴 ⊆ V
6 f1ss 6019 . . . . . . . . . . 11 ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
75, 6mpan2 703 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω–1-1→V)
87ad2antrr 758 . . . . . . . . 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 6014 . . . . . . . . . . . 12 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω⟶𝒫 𝐴)
10 frn 5966 . . . . . . . . . . . 12 (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11 uniss 4394 . . . . . . . . . . . 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 4845 . . . . . . . . . . 11 𝒫 𝐴 = 𝐴
1412, 13syl6sseq 3614 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴 ran 𝑏𝐴)
1514ad2antrr 758 . . . . . . . . 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 6009 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17 rneq 5272 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
1817unieqd 4382 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran 𝑑 = ran 𝑒)
1918sseq1d 3595 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran 𝑑𝐴 ran 𝑒𝐴))
2016, 19anbi12d 743 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) ↔ (𝑒:ω–1-1→V ∧ ran 𝑒𝐴)))
21 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (𝑐𝑑) = (𝑐𝑒))
22 f1eq1 6009 . . . . . . . . . . . . . . 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 5274 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran (𝑐𝑑) = ran (𝑐𝑒))
2524unieqd 4382 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran (𝑐𝑑) = ran (𝑐𝑒))
2625, 18psseq12d 3663 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran (𝑐𝑑) ⊊ ran 𝑑 ran (𝑐𝑒) ⊊ ran 𝑒))
2723, 26anbi12d 743 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → (((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑) ↔ ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
2820, 27imbi12d 333 . . . . . . . . . . . 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))))
2928cbvalv 2261 . . . . . . . . . . 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 205 . . . . . . . . . 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 2610 . . . . . . . . 9 (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
332, 8, 15, 31, 32fin23lem39 9055 . . . . . . . 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 1850 . . . . . . 7 ((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) → ¬ 𝐴𝐹)
3534pm2.01da 457 . . . . . 6 (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
3635exlimiv 1845 . . . . 5 (∃𝑏 𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
371, 36syl 17 . . . 4 (ω ≼ 𝒫 𝐴 → ¬ 𝐴𝐹)
3837con2i 133 . . 3 (𝐴𝐹 → ¬ ω ≼ 𝒫 𝐴)
39 pwexg 4776 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
40 isfin4-2 9019 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4139, 40syl 17 . . 3 (𝐴𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4238, 41mpbird 246 . 2 (𝐴𝐹 → 𝒫 𝐴 ∈ FinIV)
43 isfin3 9001 . 2 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
4442, 43sylibr 223 1 (𝐴𝐹𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  Vcvv 3173  wss 3540  wpss 3541  𝒫 cpw 4108   cuni 4372   cint 4410   class class class wbr 4583  ran crn 5039  cres 5040  suc csuc 5642  wf 5800  1-1wf1 5801  cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392  𝑚 cmap 7744  cdom 7839  FinIVcfin4 8985  FinIIIcfin3 8986
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-rep 4699  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-rmo 2904  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-se 4998  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-fin4 8992  df-fin3 8993
This theorem is referenced by:  isf33lem  9071
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