Theorem alephsing 10268

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 10100). 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 10057	. . . . . . 7 ⊢ ℵ Fn On
2		fnfun 6647	. . . . . . 7 ⊢ (ℵ Fn On → Fun ℵ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ℵ
4		simpl 484	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5		resfunexg 7214	. . . . . 6 ⊢ ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
6	3, 4, 5	sylancr 588	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7		limelon 6426	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8		onss 7769	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10		fnssres 6671	. . . . . . 7 ⊢ ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
11	1, 9, 10	sylancr 588	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12		fvres 6908	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
13	12	adantl 483	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14		alephord2i 10069	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
15	14	imp 408	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
16	13, 15	eqeltrd 2834	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
17	7, 16	sylan 581	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
18	17	ralrimiva 3147	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19		fnfvrnss 7117	. . . . . . 7 ⊢ (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
20	11, 18, 19	syl2anc 585	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21		df-f 6545	. . . . . 6 ⊢ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
22	11, 20, 21	sylanbrc 584	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23		alephsmo 10094	. . . . . 6 ⊢ Smo ℵ
24	1	fndmi 6651	. . . . . . 7 ⊢ dom ℵ = On
25	7, 24	eleqtrrdi 2845	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26		smores 8349	. . . . . 6 ⊢ ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
27	23, 25, 26	sylancr 588	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28		alephlim 10059	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
29	28	eleq2d 2820	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
30		eliun 5001	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦))
31		alephon 10061	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
32	31	onelssi 6477	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
33	32	reximi 3085	. . . . . . . 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 253	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
36	35	ralrimiv 3146	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
37		feq1 6696	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38		smoeq 8347	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39		fveq1 6888	. . . . . . . . . . . 12 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓‘𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
40	39, 12	sylan9eq 2793	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) = (ℵ‘𝑦))
41	40	sseq2d 4014	. . . . . . . . . 10 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
42	41	rexbidva 3177	. . . . . . . . 9 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
43	42	ralbidv 3178	. . . . . . . 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 1437	. . . . . . 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 3588	. . . . . 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 408	. . . . 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 1373	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)))
48		alephon 10061	. . . . 5 ⊢ (ℵ‘𝐴) ∈ On
49		cfcof 10266	. . . . 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 588	. . . 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 415	. 2 ⊢ (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53		cf0 10243	. . 3 ⊢ (cf‘∅) = ∅
54		fvprc 6881	. . . 4 ⊢ (¬ 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
55	54	fveq2d 6893	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56		fvprc 6881	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘𝐴) = ∅)
57	53, 55, 56	3eqtr4a 2799	. 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 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3948  ∅c0 4322  ∪ ciun 4997  dom cdm 5676  ran crn 5677   ↾ cres 5678  Oncon0 6362  Lim wlim 6363  Fun wfun 6535   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  Smo wsmo 8342  ℵcale 9928  cfccf 9929

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-smo 8343  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-oi 9502  df-har 9549  df-card 9931  df-aleph 9932  df-cf 9933  df-acn 9934

This theorem is referenced by:  alephom  10577  winafp  10689