Theorem cfub 10202

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 10200	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		dfss3 3935	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦)
3		ssel 3940	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
4		onelon 6357	. . . . . . . . . . . . . . . . 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 6374	. . . . . . . . . . . . . . 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 1917	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦) → ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
12		eluni 4874	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13		df-rex 3054	. . . . . . . . . . 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 3147	. . . . . . . . 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 1917	. . . . 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 4029	. . . 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 4933	. . . 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 3981	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
24		cff 10201	. . . . . 6 ⊢ cf:On⟶On
25	24	fdmi 6699	. . . . 5 ⊢ dom cf = On
26	25	eleq2i 2820	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27		ndmfv 6893	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
28	26, 27	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
29		0ss 4363	. . 3 ⊢ ∅ ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
30	28, 29	eqsstrdi 3991	. 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 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296  ∪ cuni 4871  ∩ cint 4910  dom cdm 5638  Oncon0 6332  ‘cfv 6511  cardccrd 9888  cfccf 9890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-card 9892  df-cf 9894

This theorem is referenced by:  cflm  10203  cf0  10204