Theorem cflm 10163

Description: Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)

Assertion

Ref	Expression
cflm	⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem cflm

Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3452	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		limsuc 7789	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝐴 → (𝑣 ∈ 𝐴 ↔ suc 𝑣 ∈ 𝐴))
3	2	biimpd 230	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝐴 → (𝑣 ∈ 𝐴 → suc 𝑣 ∈ 𝐴))
4		sseq1 3940	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = suc 𝑣 → (𝑧 ⊆ 𝑤 ↔ suc 𝑣 ⊆ 𝑤))
5	4	rexbidv 3163	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc 𝑣 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤))
6	5	rspcv 3556	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑣 ∈ 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → ∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤))
7		sucssel 6407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ V → (suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑤))
8	7	elv 3436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑤)
9	8	reximi 3077	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤 → ∃𝑤 ∈ 𝑦 𝑣 ∈ 𝑤)
10		eluni2 4842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ∪ 𝑦 ↔ ∃𝑤 ∈ 𝑦 𝑣 ∈ 𝑤)
11	9, 10	sylibr 235	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ ∪ 𝑦)
12	6, 11	syl6com 37	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (suc 𝑣 ∈ 𝐴 → 𝑣 ∈ ∪ 𝑦))
13	3, 12	syl9 77	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (𝑣 ∈ 𝐴 → 𝑣 ∈ ∪ 𝑦)))
14	13	ralrimdv 3137	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → ∀𝑣 ∈ 𝐴 𝑣 ∈ ∪ 𝑦))
15		dfss3 3904	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑣 ∈ 𝐴 𝑣 ∈ ∪ 𝑦)
16	14, 15	imbitrrdi 253	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 ⊆ ∪ 𝑦))
17	16	adantr 481	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 ⊆ ∪ 𝑦))
18		uniss 4846	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝐴 → ∪ 𝑦 ⊆ ∪ 𝐴)
19		limuni 6372	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
20	19	sseq2d 3947	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → (∪ 𝑦 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ ∪ 𝐴))
21	18, 20	imbitrrid 247	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝑦 ⊆ 𝐴 → ∪ 𝑦 ⊆ 𝐴))
22	21	imp 407	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → ∪ 𝑦 ⊆ 𝐴)
23	17, 22	jctird 531	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (𝐴 ⊆ ∪ 𝑦 ∧ ∪ 𝑦 ⊆ 𝐴)))
24		eqss 3930	. . . . . . . . . . . 12 ⊢ (𝐴 = ∪ 𝑦 ↔ (𝐴 ⊆ ∪ 𝑦 ∧ ∪ 𝑦 ⊆ 𝐴))
25	23, 24	imbitrrdi 253	. . . . . . . . . . 11 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 = ∪ 𝑦))
26	25	imdistanda 576	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) → (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)))
27	26	anim2d 618	. . . . . . . . 9 ⊢ (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))))
28	27	eximdv 1924	. . . . . . . 8 ⊢ (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))))
29	28	ss2abdv 3996	. . . . . . 7 ⊢ (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
30		intss 4899	. . . . . . 7 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
31	29, 30	syl 17	. . . . . 6 ⊢ (Lim 𝐴 → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
32	31	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
33		limelon 6375	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
34		cfval 10160	. . . . . 6 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
35	33, 34	syl 17	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
36	32, 35	sseqtrrd 3952	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴))
37		cfub 10162	. . . . 5 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
38		eqimss 3973	. . . . . . . . . 10 ⊢ (𝐴 = ∪ 𝑦 → 𝐴 ⊆ ∪ 𝑦)
39	38	anim2i 623	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦) → (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))
40	39	anim2i 623	. . . . . . . 8 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
41	40	eximi 1842	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
42	41	ss2abi 3997	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
43		intss 4899	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
44	42, 43	ax-mp 5	. . . . 5 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))}
45	37, 44	sstri 3924	. . . 4 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))}
46	36, 45	jctil 524	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ∧ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴)))
47		eqss 3930	. . 3 ⊢ ((cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ↔ ((cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ∧ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴)))
48	46, 47	sylibr 235	. 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
49	1, 48	sylan 586	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∪ cuni 4838  ∩ cint 4877  Oncon0 6310  Lim wlim 6311  suc csuc 6312  ‘cfv 6485  cardccrd 9850  cfccf 9852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-card 9854  df-cf 9856

This theorem is referenced by:  gruina  10732