Theorem fin23lem41 9375

Description: Lemma for fin23 9412. 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 8119	. . . . 5 ⊢ (ω ≼ 𝒫 𝐴 → ∃𝑏 𝑏:ω–1-1→𝒫 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 9368	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐹 → ∃𝑐∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)))
4	3	adantl 467	. . . . . . . 8 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ∃𝑐∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)))
5		ssv 3772	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ⊆ V
6		f1ss 6246	. . . . . . . . . . 11 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
7	5, 6	mpan2 663	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω–1-1→V)
8	7	ad2antrr 697	. . . . . . . . 9 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → 𝑏:ω–1-1→V)
9		f1f 6241	. . . . . . . . . . . 12 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω⟶𝒫 𝐴)
10		frn 6193	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11		uniss 4593	. . . . . . . . . . . 12 ⊢ (ran 𝑏 ⊆ 𝒫 𝐴 → ∪ ran 𝑏 ⊆ ∪ 𝒫 𝐴)
12	9, 10, 11	3syl 18	. . . . . . . . . . 11 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ ∪ 𝒫 𝐴)
13		unipw 5046	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐴 = 𝐴
14	12, 13	syl6sseq 3798	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ 𝐴)
15	14	ad2antrr 697	. . . . . . . . 9 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ∪ ran 𝑏 ⊆ 𝐴)
16		f1eq1 6236	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17		rneq 5489	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
18	17	unieqd 4582	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran 𝑑 = ∪ ran 𝑒)
19	18	sseq1d 3779	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran 𝑑 ⊆ 𝐴 ↔ ∪ ran 𝑒 ⊆ 𝐴))
20	16, 19	anbi12d 608	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) ↔ (𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴)))
21		fveq2 6332	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (𝑐‘𝑑) = (𝑐‘𝑒))
22		f1eq1 6236	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑑) = (𝑐‘𝑒) → ((𝑐‘𝑑):ω–1-1→V ↔ (𝑐‘𝑒):ω–1-1→V))
23	21, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → ((𝑐‘𝑑):ω–1-1→V ↔ (𝑐‘𝑒):ω–1-1→V))
24	21	rneqd 5491	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran (𝑐‘𝑑) = ran (𝑐‘𝑒))
25	24	unieqd 4582	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran (𝑐‘𝑑) = ∪ ran (𝑐‘𝑒))
26	25, 18	psseq12d 3849	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑 ↔ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))
27	23, 26	anbi12d 608	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑) ↔ ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
28	20, 27	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → (((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) ↔ ((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))))
29	28	cbvalv 2433	. . . . . . . . . . 11 ⊢ (∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) ↔ ∀𝑒((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
30	29	biimpi 206	. . . . . . . . . 10 ⊢ (∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) → ∀𝑒((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
31	30	adantl 467	. . . . . . . . 9 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ∀𝑒((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
32		eqid 2770	. . . . . . . . 9 ⊢ (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
33	2, 8, 15, 31, 32	fin23lem39 9373	. . . . . . . 8 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ¬ 𝐴 ∈ 𝐹)
34	4, 33	exlimddv 2014	. . . . . . 7 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ¬ 𝐴 ∈ 𝐹)
35	34	pm2.01da 792	. . . . . 6 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
36	35	exlimiv 2009	. . . . 5 ⊢ (∃𝑏 𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
37	1, 36	syl 17	. . . 4 ⊢ (ω ≼ 𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
38	37	con2i 136	. . 3 ⊢ (𝐴 ∈ 𝐹 → ¬ ω ≼ 𝒫 𝐴)
39		pwexg 4978	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
40		isfin4-2 9337	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
41	39, 40	syl 17	. . 3 ⊢ (𝐴 ∈ 𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
42	38, 41	mpbird 247	. 2 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ FinIV)
43		isfin3 9319	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
44	42, 43	sylibr 224	1 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1628   = wceq 1630  ∃wex 1851   ∈ wcel 2144  {cab 2756  ∀wral 3060  Vcvv 3349   ⊆ wss 3721   ⊊ wpss 3722  𝒫 cpw 4295  ∪ cuni 4572  ∩ cint 4609   class class class wbr 4784  ran crn 5250   ↾ cres 5251  suc csuc 5868  ⟶wf 6027  –1-1→wf1 6028  ‘cfv 6031  (class class class)co 6792  ωcom 7211  reccrdg 7657   ↑_𝑚 cmap 8008   ≼ cdom 8106  Fin^IVcfin4 9303  Fin^IIIcfin3 9304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-fin4 9310  df-fin3 9311

This theorem is referenced by:  isf33lem  9389