Theorem alephsing 10236

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 10068). 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 10025	. . . . . . 7 ⊢ ℵ Fn On
2		fnfun 6621	. . . . . . 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 7192	. . . . . 6 ⊢ ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
6	3, 4, 5	sylancr 587	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7		limelon 6400	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8		onss 7764	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10		fnssres 6644	. . . . . . 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 6880	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
13	12	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14		alephord2i 10037	. . . . . . . . . . 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 2829	. . . . . . . . 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 3126	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19		fnfvrnss 7096	. . . . . . 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 6518	. . . . . 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 10062	. . . . . 6 ⊢ Smo ℵ
24	1	fndmi 6625	. . . . . . 7 ⊢ dom ℵ = On
25	7, 24	eleqtrrdi 2840	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26		smores 8324	. . . . . 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 10027	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
29	28	eleq2d 2815	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
30		eliun 4962	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦))
31		alephon 10029	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
32	31	onelssi 6452	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
33	32	reximi 3068	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
34	30, 33	sylbi 217	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
35	29, 34	biimtrdi 253	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
36	35	ralrimiv 3125	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
37		feq1 6669	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38		smoeq 8322	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39		fveq1 6860	. . . . . . . . . . . 12 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓‘𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
40	39, 12	sylan9eq 2785	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) = (ℵ‘𝑦))
41	40	sseq2d 3982	. . . . . . . . . 10 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
42	41	rexbidva 3156	. . . . . . . . 9 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
43	42	ralbidv 3157	. . . . . . . 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 1438	. . . . . . 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 3566	. . . . . 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 1374	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)))
48		alephon 10029	. . . . 5 ⊢ (ℵ‘𝐴) ∈ On
49		cfcof 10234	. . . . 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 413	. 2 ⊢ (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53		cf0 10211	. . 3 ⊢ (cf‘∅) = ∅
54		fvprc 6853	. . . 4 ⊢ (¬ 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
55	54	fveq2d 6865	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56		fvprc 6853	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘𝐴) = ∅)
57	53, 55, 56	3eqtr4a 2791	. 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 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917  ∅c0 4299  ∪ ciun 4958  dom cdm 5641  ran crn 5642   ↾ cres 5643  Oncon0 6335  Lim wlim 6336  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  Smo wsmo 8317  ℵcale 9896  cfccf 9897

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-smo 8318  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-oi 9470  df-har 9517  df-card 9899  df-aleph 9900  df-cf 9901  df-acn 9902

This theorem is referenced by:  alephom  10545  winafp  10657