Theorem alephsing 10032

Description: The cofinality of a limit aleph is the same as the cofinality of its argument, so if (ℵ‘𝐴) < 𝐴, then (ℵ‘𝐴) is singular. Conversely, if (ℵ‘𝐴) is regular (i.e. weakly inaccessible), then (ℵ‘𝐴) = 𝐴, so 𝐴 has to be rather large (see alephfp 9864). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

Assertion

Ref	Expression
alephsing	⊢ (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))

Proof of Theorem alephsing

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephfnon 9821	. . . . . . 7 ⊢ ℵ Fn On
2		fnfun 6533	. . . . . . 7 ⊢ (ℵ Fn On → Fun ℵ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ℵ
4		simpl 483	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5		resfunexg 7091	. . . . . 6 ⊢ ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
6	3, 4, 5	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7		limelon 6329	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8		onss 7634	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10		fnssres 6555	. . . . . . 7 ⊢ ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
11	1, 9, 10	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12		fvres 6793	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
13	12	adantl 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14		alephord2i 9833	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
15	14	imp 407	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
16	13, 15	eqeltrd 2839	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
17	7, 16	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
18	17	ralrimiva 3103	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19		fnfvrnss 6994	. . . . . . 7 ⊢ (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
20	11, 18, 19	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21		df-f 6437	. . . . . 6 ⊢ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
22	11, 20, 21	sylanbrc 583	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23		alephsmo 9858	. . . . . 6 ⊢ Smo ℵ
24	1	fndmi 6537	. . . . . . 7 ⊢ dom ℵ = On
25	7, 24	eleqtrrdi 2850	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26		smores 8183	. . . . . 6 ⊢ ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
27	23, 25, 26	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28		alephlim 9823	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
29	28	eleq2d 2824	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
30		eliun 4928	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦))
31		alephon 9825	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
32	31	onelssi 6375	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
33	32	reximi 3178	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
34	30, 33	sylbi 216	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
35	29, 34	syl6bi 252	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
36	35	ralrimiv 3102	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
37		feq1 6581	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38		smoeq 8181	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39		fveq1 6773	. . . . . . . . . . . 12 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓‘𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
40	39, 12	sylan9eq 2798	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) = (ℵ‘𝑦))
41	40	sseq2d 3953	. . . . . . . . . 10 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
42	41	rexbidva 3225	. . . . . . . . 9 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
43	42	ralbidv 3112	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
44	37, 38, 43	3anbi123d 1435	. . . . . . 7 ⊢ (𝑓 = (ℵ ↾ 𝐴) → ((𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)) ↔ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ∧ Smo (ℵ ↾ 𝐴) ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))))
45	44	spcegv 3536	. . . . . 6 ⊢ ((ℵ ↾ 𝐴) ∈ V → (((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ∧ Smo (ℵ ↾ 𝐴) ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦))))
46	45	imp 407	. . . . 5 ⊢ (((ℵ ↾ 𝐴) ∈ V ∧ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ∧ Smo (ℵ ↾ 𝐴) ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)))
47	6, 22, 27, 36, 46	syl13anc 1371	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)))
48		alephon 9825	. . . . 5 ⊢ (ℵ‘𝐴) ∈ On
49		cfcof 10030	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ On ∧ 𝐴 ∈ On) → (∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
50	48, 7, 49	sylancr 587	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
51	47, 50	mpd 15	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
52	51	expcom 414	. 2 ⊢ (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53		cf0 10007	. . 3 ⊢ (cf‘∅) = ∅
54		fvprc 6766	. . . 4 ⊢ (¬ 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
55	54	fveq2d 6778	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56		fvprc 6766	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘𝐴) = ∅)
57	53, 55, 56	3eqtr4a 2804	. 2 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
58	52, 57	pm2.61d1 180	1 ⊢ (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  ∪ ciun 4924  dom cdm 5589  ran crn 5590   ↾ cres 5591  Oncon0 6266  Lim wlim 6267  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  Smo wsmo 8176  ℵcale 9694  cfccf 9695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-smo 8177  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-har 9316  df-card 9697  df-aleph 9698  df-cf 9699  df-acn 9700

This theorem is referenced by:  alephom  10341  winafp  10453