Theorem alephsing 9963

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 9795). 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 9752	. . . . . . 7 ⊢ ℵ Fn On
2		fnfun 6517	. . . . . . 7 ⊢ (ℵ Fn On → Fun ℵ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ℵ
4		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5		resfunexg 7073	. . . . . 6 ⊢ ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
6	3, 4, 5	sylancr 586	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7		limelon 6314	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8		onss 7611	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10		fnssres 6539	. . . . . . 7 ⊢ ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
11	1, 9, 10	sylancr 586	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12		fvres 6775	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
13	12	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14		alephord2i 9764	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
15	14	imp 406	. . . . . . . . . 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 579	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
18	17	ralrimiva 3107	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19		fnfvrnss 6976	. . . . . . 7 ⊢ (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
20	11, 18, 19	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21		df-f 6422	. . . . . 6 ⊢ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
22	11, 20, 21	sylanbrc 582	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23		alephsmo 9789	. . . . . 6 ⊢ Smo ℵ
24	1	fndmi 6521	. . . . . . 7 ⊢ dom ℵ = On
25	7, 24	eleqtrrdi 2850	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26		smores 8154	. . . . . 6 ⊢ ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
27	23, 25, 26	sylancr 586	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28		alephlim 9754	. . . . . . . 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 4925	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦))
31		alephon 9756	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
32	31	onelssi 6360	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
33	32	reximi 3174	. . . . . . . 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 3106	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
37		feq1 6565	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38		smoeq 8152	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39		fveq1 6755	. . . . . . . . . . . 12 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓‘𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
40	39, 12	sylan9eq 2799	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) = (ℵ‘𝑦))
41	40	sseq2d 3949	. . . . . . . . . 10 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
42	41	rexbidva 3224	. . . . . . . . 9 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
43	42	ralbidv 3120	. . . . . . . 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 1434	. . . . . . 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 3526	. . . . . 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 406	. . . . 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 1370	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)))
48		alephon 9756	. . . . 5 ⊢ (ℵ‘𝐴) ∈ On
49		cfcof 9961	. . . . 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 586	. . . 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 413	. 2 ⊢ (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53		cf0 9938	. . 3 ⊢ (cf‘∅) = ∅
54		fvprc 6748	. . . 4 ⊢ (¬ 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
55	54	fveq2d 6760	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56		fvprc 6748	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘𝐴) = ∅)
57	53, 55, 56	3eqtr4a 2805	. 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 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∪ ciun 4921  dom cdm 5580  ran crn 5581   ↾ cres 5582  Oncon0 6251  Lim wlim 6252  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  Smo wsmo 8147  ℵcale 9625  cfccf 9626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-smo 8148  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629  df-cf 9630  df-acn 9631

This theorem is referenced by:  alephom  10272  winafp  10384