Theorem cfub 10287

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 10285	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		dfss3 3984	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦)
3		ssel 3989	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
4		onelon 6411	. . . . . . . . . . . . . . . . 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 6428	. . . . . . . . . . . . . . 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 1915	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) → ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
12		eluni 4915	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13		df-rex 3069	. . . . . . . . . . 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 3167	. . . . . . . . 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 1915	. . . . 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 4076	. . . 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 4974	. . . 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 4034	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
24		cff 10286	. . . . . 6 ⊢ cf:On⟶On
25	24	fdmi 6748	. . . . 5 ⊢ dom cf = On
26	25	eleq2i 2831	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27		ndmfv 6942	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
28	26, 27	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
29		0ss 4406	. . 3 ⊢ ∅ ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
30	28, 29	eqsstrdi 4050	. 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 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ∅c0 4339  ∪ cuni 4912  ∩ cint 4951  dom cdm 5689  Oncon0 6386  ‘cfv 6563  cardccrd 9973  cfccf 9975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-card 9977  df-cf 9979

This theorem is referenced by:  cflm  10288  cf0  10289