Theorem cfub 10290

Description: An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cfub	⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem cfub

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

Step	Hyp	Ref	Expression
1		cfval 10288	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		dfss3 3971	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦)
3		ssel 3976	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
4		onelon 6408	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ On)
5	4	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → (𝑤 ∈ 𝐴 → 𝑤 ∈ On))
6	3, 5	sylan9r 508	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑤 ∈ 𝑦 → 𝑤 ∈ On))
7		onelss 6425	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ On → (𝑧 ∈ 𝑤 → 𝑧 ⊆ 𝑤))
8	6, 7	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑤 ∈ 𝑦 → (𝑧 ∈ 𝑤 → 𝑧 ⊆ 𝑤)))
9	8	imdistand 570	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → ((𝑤 ∈ 𝑦 ∧ 𝑧 ∈ 𝑤) → (𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
10	9	ancomsd 465	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) → (𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
11	10	eximdv 1916	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) → ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
12		eluni 4909	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13		df-rex 3070	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤))
14	11, 12, 13	3imtr4g 296	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑧 ∈ ∪ 𝑦 → ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
15	14	ralimdv 3168	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
16	2, 15	biimtrid 242	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝑦 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
17	16	imdistanda 571	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦) → (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
18	17	anim2d 612	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
19	18	eximdv 1916	. . . . 5 ⊢ (𝐴 ∈ On → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
20	19	ss2abdv 4065	. . . 4 ⊢ (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
21		intss 4968	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
22	20, 21	syl 17	. . 3 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
23	1, 22	eqsstrd 4017	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
24		cff 10289	. . . . . 6 ⊢ cf:On⟶On
25	24	fdmi 6746	. . . . 5 ⊢ dom cf = On
26	25	eleq2i 2832	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27		ndmfv 6940	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
28	26, 27	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
29		0ss 4399	. . 3 ⊢ ∅ ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
30	28, 29	eqsstrdi 4027	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
31	23, 30	pm2.61i 182	1 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069   ⊆ wss 3950  ∅c0 4332  ∪ cuni 4906  ∩ cint 4945  dom cdm 5684  Oncon0 6383  ‘cfv 6560  cardccrd 9976  cfccf 9978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-card 9980  df-cf 9982

This theorem is referenced by:  cflm  10291  cf0  10292