Theorem cfub 9660

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 9658	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		dfss3 3903	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦)
3		ssel 3908	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
4		onelon 6184	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ On)
5	4	ex 416	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → (𝑤 ∈ 𝐴 → 𝑤 ∈ On))
6	3, 5	sylan9r 512	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑤 ∈ 𝑦 → 𝑤 ∈ On))
7		onelss 6201	. . . . . . . . . . . . . . 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 574	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → ((𝑤 ∈ 𝑦 ∧ 𝑧 ∈ 𝑤) → (𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
10	9	ancomsd 469	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) → (𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
11	10	eximdv 1918	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) → ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
12		eluni 4803	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13		df-rex 3112	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤))
14	11, 12, 13	3imtr4g 299	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑧 ∈ ∪ 𝑦 → ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
15	14	ralimdv 3145	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
16	2, 15	syl5bi 245	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝑦 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
17	16	imdistanda 575	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦) → (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
18	17	anim2d 614	. . . . . 6 ⊢ (𝐴 ∈ On → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
19	18	eximdv 1918	. . . . 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 3991	. . . 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 4859	. . . 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 3953	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
24		cff 9659	. . . . . 6 ⊢ cf:On⟶On
25	24	fdmi 6498	. . . . 5 ⊢ dom cf = On
26	25	eleq2i 2881	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27		ndmfv 6675	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
28	26, 27	sylnbir 334	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
29		0ss 4304	. . 3 ⊢ ∅ ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
30	28, 29	eqsstrdi 3969	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
31	23, 30	pm2.61i 185	1 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881  ∅c0 4243  ∪ cuni 4800  ∩ cint 4838  dom cdm 5519  Oncon0 6159  ‘cfv 6324  cardccrd 9348  cfccf 9350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-card 9352  df-cf 9354

This theorem is referenced by:  cflm  9661  cf0  9662