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Theorem fin23lem41 10377
Description: Lemma for fin23 10414. 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 8979 . . . . 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 10370 . . . . . . . . 9 (𝐴𝐹 → ∃𝑐𝑑((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) → ((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑)))
43adantl 480 . . . . . . . 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 4001 . . . . . . . . . . 11 𝒫 𝐴 ⊆ V
6 f1ss 6798 . . . . . . . . . . 11 ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
75, 6mpan2 689 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω–1-1→V)
87ad2antrr 724 . . . . . . . . 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 6793 . . . . . . . . . . . 12 (𝑏:ω–1-1→𝒫 𝐴𝑏:ω⟶𝒫 𝐴)
10 frn 6730 . . . . . . . . . . . 12 (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11 uniss 4917 . . . . . . . . . . . 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 5452 . . . . . . . . . . 11 𝒫 𝐴 = 𝐴
1412, 13sseqtrdi 4027 . . . . . . . . . 10 (𝑏:ω–1-1→𝒫 𝐴 ran 𝑏𝐴)
1514ad2antrr 724 . . . . . . . . 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 6788 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17 rneq 5938 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
1817unieqd 4922 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran 𝑑 = ran 𝑒)
1918sseq1d 4008 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran 𝑑𝐴 ran 𝑒𝐴))
2016, 19anbi12d 630 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ran 𝑑𝐴) ↔ (𝑒:ω–1-1→V ∧ ran 𝑒𝐴)))
21 fveq2 6896 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 → (𝑐𝑑) = (𝑐𝑒))
22 f1eq1 6788 . . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . . . 16 (𝑑 = 𝑒 → ran (𝑐𝑑) = ran (𝑐𝑒))
2524unieqd 4922 . . . . . . . . . . . . . . 15 (𝑑 = 𝑒 ran (𝑐𝑑) = ran (𝑐𝑒))
2625, 18psseq12d 4090 . . . . . . . . . . . . . 14 (𝑑 = 𝑒 → ( ran (𝑐𝑑) ⊊ ran 𝑑 ran (𝑐𝑒) ⊊ ran 𝑒))
2723, 26anbi12d 630 . . . . . . . . . . . . 13 (𝑑 = 𝑒 → (((𝑐𝑑):ω–1-1→V ∧ ran (𝑐𝑑) ⊊ ran 𝑑) ↔ ((𝑐𝑒):ω–1-1→V ∧ ran (𝑐𝑒) ⊊ ran 𝑒)))
2820, 27imbi12d 343 . . . . . . . . . . . 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 2031 . . . . . . . . . . 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 215 . . . . . . . . . 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 480 . . . . . . . . 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 2725 . . . . . . . . 9 (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
332, 8, 15, 31, 32fin23lem39 10375 . . . . . . . 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 1930 . . . . . . 7 ((𝑏:ω–1-1→𝒫 𝐴𝐴𝐹) → ¬ 𝐴𝐹)
3534pm2.01da 797 . . . . . 6 (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴𝐹)
3635exlimiv 1925 . . . . 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 10339 . . . 4 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4139, 40syl 17 . . 3 (𝐴𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
4238, 41mpbird 256 . 2 (𝐴𝐹 → 𝒫 𝐴 ∈ FinIV)
43 isfin3 10321 . 2 (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
4442, 43sylibr 233 1 (𝐴𝐹𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wral 3050  Vcvv 3461  wss 3944  wpss 3945  𝒫 cpw 4604   cuni 4909   cint 4950   class class class wbr 5149  ran crn 5679  cres 5680  suc csuc 6373  wf 6545  1-1wf1 6546  cfv 6549  (class class class)co 7419  ωcom 7871  reccrdg 8430  m cmap 8845  cdom 8962  FinIVcfin4 10305  FinIIIcfin3 10306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-seqom 8469  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9964  df-fin4 10312  df-fin3 10313
This theorem is referenced by:  isf33lem  10391
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