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Theorem fin23lem41 9159
Description: Lemma for fin23 9196. 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 7951 . . . . 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 9152 . . . . . . . . 9 (𝐴𝐹 → ∃𝑐𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)))
43adantl 482 . . . . . . . 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 3617 . . . . . . . . . . 11 𝒫 𝐴 ⊆ V
6 f1ss 6093 . . . . . . . . . . 11 ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
75, 6mpan2 706 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω–1-1→V)
87ad2antrr 761 . . . . . . . . 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 6088 . . . . . . . . . . . 12 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω⟶𝒫 𝐴)
10 frn 6040 . . . . . . . . . . . 12 (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11 uniss 4449 . . . . . . . . . . . 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 4909 . . . . . . . . . . 11 𝒫 𝐴 = 𝐴
1412, 13syl6sseq 3643 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴 ran 𝑏𝐴)
1514ad2antrr 761 . . . . . . . . 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 6083 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17 rneq 5340 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
1817unieqd 4437 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran 𝑑 = ran 𝑒)
1918sseq1d 3624 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran 𝑑𝐴 ran 𝑒𝐴))
2016, 19anbi12d 746 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) ↔ (𝑒:ω–1-1→V ∧ ran 𝑒𝐴)))
21 fveq2 6178 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (𝑐𝑑) = (𝑐𝑒))
22 f1eq1 6083 . . . . . . . . . . . . . . 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 5342 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran (𝑐𝑑) = ran (𝑐𝑒))
2524unieqd 4437 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran (𝑐𝑑) = ran (𝑐𝑒))
2625, 18psseq12d 3693 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran (𝑐𝑑) ⊊ ran 𝑑 ran (𝑐𝑒) ⊊ ran 𝑒))
2723, 26anbi12d 746 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → (((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑) ↔ ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
2820, 27imbi12d 334 . . . . . . . . . . . 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 2271 . . . . . . . . . . 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 206 . . . . . . . . . 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 482 . . . . . . . . 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 2620 . . . . . . . . 9 (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
332, 8, 15, 31, 32fin23lem39 9157 . . . . . . . 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 1861 . . . . . . 7 ((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) → ¬ 𝐴𝐹)
3534pm2.01da 458 . . . . . 6 (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
3635exlimiv 1856 . . . . 5 (∃𝑏 𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
371, 36syl 17 . . . 4 (ω ≼ 𝒫 𝐴 → ¬ 𝐴𝐹)
3837con2i 134 . . 3 (𝐴𝐹 → ¬ ω ≼ 𝒫 𝐴)
39 pwexg 4841 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
40 isfin4-2 9121 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4139, 40syl 17 . . 3 (𝐴𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4238, 41mpbird 247 . 2 (𝐴𝐹 → 𝒫 𝐴 ∈ FinIV)
43 isfin3 9103 . 2 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
4442, 43sylibr 224 1 (𝐴𝐹𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  Vcvv 3195  wss 3567  wpss 3568  𝒫 cpw 4149   cuni 4427   cint 4466   class class class wbr 4644  ran crn 5105  cres 5106  suc csuc 5713  wf 5872  1-1wf1 5873  cfv 5876  (class class class)co 6635  ωcom 7050  reccrdg 7490  𝑚 cmap 7842  cdom 7938  FinIVcfin4 9087  FinIIIcfin3 9088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-fin4 9094  df-fin3 9095
This theorem is referenced by:  isf33lem  9173
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