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Theorem fin23lem41 9118
Description: Lemma for fin23 9155. 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 7910 . . . . 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 9111 . . . . . . . . 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 3604 . . . . . . . . . . 11 𝒫 𝐴 ⊆ V
6 f1ss 6063 . . . . . . . . . . 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 6058 . . . . . . . . . . . 12 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω⟶𝒫 𝐴)
10 frn 6010 . . . . . . . . . . . 12 (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11 uniss 4424 . . . . . . . . . . . 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 4879 . . . . . . . . . . 11 𝒫 𝐴 = 𝐴
1412, 13syl6sseq 3630 . . . . . . . . . 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 6053 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17 rneq 5311 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
1817unieqd 4412 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran 𝑑 = ran 𝑒)
1918sseq1d 3611 . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (𝑐𝑑) = (𝑐𝑒))
22 f1eq1 6053 . . . . . . . . . . . . . . 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 5313 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran (𝑐𝑑) = ran (𝑐𝑒))
2524unieqd 4412 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran (𝑐𝑑) = ran (𝑐𝑒))
2625, 18psseq12d 3679 . . . . . . . . . . . . . 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 2272 . . . . . . . . . . 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 2621 . . . . . . . . 9 (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
332, 8, 15, 31, 32fin23lem39 9116 . . . . . . . 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 1860 . . . . . . 7 ((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) → ¬ 𝐴𝐹)
3534pm2.01da 458 . . . . . 6 (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
3635exlimiv 1855 . . . . 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 4810 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
40 isfin4-2 9080 . . . 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 9062 . 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 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  wss 3555  wpss 3556  𝒫 cpw 4130   cuni 4402   cint 4440   class class class wbr 4613  ran crn 5075  cres 5076  suc csuc 5684  wf 5843  1-1wf1 5844  cfv 5847  (class class class)co 6604  ωcom 7012  reccrdg 7450  𝑚 cmap 7802  cdom 7897  FinIVcfin4 9046  FinIIIcfin3 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-fin4 9053  df-fin3 9054
This theorem is referenced by:  isf33lem  9132
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