Theorem fin23lem41 10454

Description: Lemma for fin23 10491. 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 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
fin23lem41	⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ 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.

Step	Hyp	Ref	Expression
1		brdomi 9011	. . . . 5 ⊢ (ω ≼ 𝒫 𝐴 → ∃𝑏 𝑏:ω⟶1-1𝒫 𝐴)
2		fin23lem40.f	. . . . . . . . . 10 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
3	2	fin23lem33 10447	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐹 → ∃𝑐∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)))
4	3	adantl 487	. . . . . . . 8 ⊢ ((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ∃𝑐∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)))
5		ssv 3937	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ⊆ V
6		f1ss 6804	. . . . . . . . . . 11 ⊢ ((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω⟶1-1V)
7	5, 6	mpan2 704	. . . . . . . . . 10 ⊢ (𝑏:ω⟶1-1𝒫 𝐴 → 𝑏:ω⟶1-1V)
8	7	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → 𝑏:ω⟶1-1V)
9		f1f 6797	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶1-1𝒫 𝐴 → 𝑏:ω⟶𝒫 𝐴)
10		frn 6736	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11		uniss 4878	. . . . . . . . . . . 12 ⊢ (ran 𝑏 ⊆ 𝒫 𝐴 → ∪ ran 𝑏 ⊆ ∪ 𝒫 𝐴)
12	9, 10, 11	3syl 19	. . . . . . . . . . 11 ⊢ (𝑏:ω⟶1-1𝒫 𝐴 → ∪ ran 𝑏 ⊆ ∪ 𝒫 𝐴)
13		unipw 5422	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐴 = 𝐴
14	12, 13	sseqtrdi 3953	. . . . . . . . . 10 ⊢ (𝑏:ω⟶1-1𝒫 𝐴 → ∪ ran 𝑏 ⊆ 𝐴)
15	14	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ∪ ran 𝑏 ⊆ 𝐴)
16		f1eq1 6792	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (𝑑:ω⟶1-1V ↔ 𝑒:ω⟶1-1V))
17		rneq 5924	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
18	17	unieqd 4883	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → ∪ ran 𝑑 = ∪ ran 𝑒)
19	18	sseq1d 3944	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (∪ ran 𝑑 ⊆ 𝐴 ↔ ∪ ran 𝑒 ⊆ 𝐴))
20	16, 19	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → ((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) ↔ (𝑒:ω⟶1-1V ∧ ∪ ran 𝑒 ⊆ 𝐴)))
21		fveq2 6905	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (𝑐‘𝑑) = (𝑐‘𝑒))
22		f1eq1 6792	. . . . . . . . . . . . . 14 ⊢ ((𝑐‘𝑑) = (𝑐‘𝑒) → ((𝑐‘𝑑):ω⟶1-1V ↔ (𝑐‘𝑒):ω⟶1-1V))
23	21, 22	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → ((𝑐‘𝑑):ω⟶1-1V ↔ (𝑐‘𝑒):ω⟶1-1V))
24	21	rneqd 5926	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ran (𝑐‘𝑑) = ran (𝑐‘𝑒))
25	24	unieqd 4883	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → ∪ ran (𝑐‘𝑑) = ∪ ran (𝑐‘𝑒))
26	25, 18	psseq12d 4027	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑 ↔ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))
27	23, 26	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → (((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑) ↔ ((𝑐‘𝑒):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
28	20, 27	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑒 → (((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) ↔ ((𝑒:ω⟶1-1V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))))
29	28	cbvalvw 2071	. . . . . . . . . 10 ⊢ (∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) ↔ ∀𝑒((𝑒:ω⟶1-1V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
30	29	bilani 510	. . . . . . . . 9 ⊢ (((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ∀𝑒((𝑒:ω⟶1-1V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
31		eqid 2763	. . . . . . . . 9 ⊢ (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
32	2, 8, 15, 30, 31	fin23lem39 10452	. . . . . . . 8 ⊢ (((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω⟶1-1V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω⟶1-1V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ¬ 𝐴 ∈ 𝐹)
33	4, 32	exlimddv 1970	. . . . . . 7 ⊢ ((𝑏:ω⟶1-1𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ¬ 𝐴 ∈ 𝐹)
34	33	pm2.01da 811	. . . . . 6 ⊢ (𝑏:ω⟶1-1𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
35	34	exlimiv 1965	. . . . 5 ⊢ (∃𝑏 𝑏:ω⟶1-1𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
36	1, 35	syl 18	. . . 4 ⊢ (ω ≼ 𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
37	36	con2i 140	. . 3 ⊢ (𝐴 ∈ 𝐹 → ¬ ω ≼ 𝒫 𝐴)
38		pwexg 5344	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
39		isfin4-2 10416	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
40	38, 39	syl 18	. . 3 ⊢ (𝐴 ∈ 𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
41	37, 40	mpbird 260	. 2 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ FinIV)
42		isfin3 10398	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
43	41, 42	sylibr 237	1 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910   class class class wbr 5106  ran crn 5656   ↾ cres 5657  suc csuc 6379  ⟶wf 6551  ⟶1-1wf1 6552  ‘cfv 6555  (class class class)co 7440  ωcom 7898  reccrdg 8442   ↑_m cmap 8872   ≼ cdom 8996  Fin^IVcfin4 10382  Fin^IIIcfin3 10383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-seqom 8483  df-1o 8501  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-card 10044  df-fin4 10389  df-fin3 10390

This theorem is used by:  isf33lem  10468