Theorem cflm 10141

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 3457	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		limsuc 7779	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝐴 → (𝑣 ∈ 𝐴 ↔ suc 𝑣 ∈ 𝐴))
3	2	biimpd 229	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝐴 → (𝑣 ∈ 𝐴 → suc 𝑣 ∈ 𝐴))
4		sseq1 3960	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = suc 𝑣 → (𝑧 ⊆ 𝑤 ↔ suc 𝑣 ⊆ 𝑤))
5	4	rexbidv 3156	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc 𝑣 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤))
6	5	rspcv 3573	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑣 ∈ 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → ∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤))
7		sucssel 6403	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ V → (suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑤))
8	7	elv 3441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑤)
9	8	reximi 3070	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤 → ∃𝑤 ∈ 𝑦 𝑣 ∈ 𝑤)
10		eluni2 4863	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ∪ 𝑦 ↔ ∃𝑤 ∈ 𝑦 𝑣 ∈ 𝑤)
11	9, 10	sylibr 234	. . . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → ∀𝑣 ∈ 𝐴 𝑣 ∈ ∪ 𝑦))
15		dfss3 3923	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑣 ∈ 𝐴 𝑣 ∈ ∪ 𝑦)
16	14, 15	imbitrrdi 252	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 ⊆ ∪ 𝑦))
17	16	adantr 480	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 ⊆ ∪ 𝑦))
18		uniss 4867	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝐴 → ∪ 𝑦 ⊆ ∪ 𝐴)
19		limuni 6368	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
20	19	sseq2d 3967	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → (∪ 𝑦 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ ∪ 𝐴))
21	18, 20	imbitrrid 246	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝑦 ⊆ 𝐴 → ∪ 𝑦 ⊆ 𝐴))
22	21	imp 406	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → ∪ 𝑦 ⊆ 𝐴)
23	17, 22	jctird 526	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (𝐴 ⊆ ∪ 𝑦 ∧ ∪ 𝑦 ⊆ 𝐴)))
24		eqss 3950	. . . . . . . . . . . 12 ⊢ (𝐴 = ∪ 𝑦 ↔ (𝐴 ⊆ ∪ 𝑦 ∧ ∪ 𝑦 ⊆ 𝐴))
25	23, 24	imbitrrdi 252	. . . . . . . . . . 11 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 = ∪ 𝑦))
26	25	imdistanda 571	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) → (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)))
27	26	anim2d 612	. . . . . . . . 9 ⊢ (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))))
28	27	eximdv 1918	. . . . . . . 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 4017	. . . . . . 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 4919	. . . . . . 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 481	. . . . 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 6371	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
34		cfval 10138	. . . . . 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 3972	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴))
37		cfub 10140	. . . . 5 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
38		eqimss 3993	. . . . . . . . . 10 ⊢ (𝐴 = ∪ 𝑦 → 𝐴 ⊆ ∪ 𝑦)
39	38	anim2i 617	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦) → (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))
40	39	anim2i 617	. . . . . . . 8 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
41	40	eximi 1836	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
42	41	ss2abi 4018	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
43		intss 4919	. . . . . 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 3944	. . . 4 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))}
46	36, 45	jctil 519	. . 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 3950	. . 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 234	. 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
49	1, 48	sylan 580	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3902  ∪ cuni 4859  ∩ cint 4897  Oncon0 6306  Lim wlim 6307  suc csuc 6308  ‘cfv 6481  cardccrd 9828  cfccf 9830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-card 9832  df-cf 9834

This theorem is referenced by:  gruina  10709