Theorem cfub 9674

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 9672	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		dfss3 3959	. . . . . . . . 9 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦)
3		ssel 3964	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
4		onelon 6219	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ On)
5	4	ex 415	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → (𝑤 ∈ 𝐴 → 𝑤 ∈ On))
6	3, 5	sylan9r 511	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑤 ∈ 𝑦 → 𝑤 ∈ On))
7		onelss 6236	. . . . . . . . . . . . . . 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 573	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → ((𝑤 ∈ 𝑦 ∧ 𝑧 ∈ 𝑤) → (𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤)))
10	9	ancomsd 468	. . . . . . . . . . . 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 4844	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))
13		df-rex 3147	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤(𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤))
14	11, 12, 13	3imtr4g 298	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝑧 ∈ ∪ 𝑦 → ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
15	14	ralimdv 3181	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
16	2, 15	syl5bi 244	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝑦 → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
17	16	imdistanda 574	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦) → (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
18	17	anim2d 613	. . . . . 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 4047	. . . 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 4900	. . . 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 4008	. 2 ⊢ (𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
24		cff 9673	. . . . . 6 ⊢ cf:On⟶On
25	24	fdmi 6527	. . . . 5 ⊢ dom cf = On
26	25	eleq2i 2907	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
27		ndmfv 6703	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
28	26, 27	sylnbir 333	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
29		0ss 4353	. . 3 ⊢ ∅ ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
30	28, 29	eqsstrdi 4024	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))})
31	23, 30	pm2.61i 184	1 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2802  ∀wral 3141  ∃wrex 3142   ⊆ wss 3939  ∅c0 4294  ∪ cuni 4841  ∩ cint 4879  dom cdm 5558  Oncon0 6194  ‘cfv 6358  cardccrd 9367  cfccf 9369

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ord 6197  df-on 6198  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-card 9371  df-cf 9373

This theorem is referenced by:  cflm  9675  cf0  9676