Theorem alephsing 8981

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 8814). 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 8771	. . . . . . 7 ⊢ ℵ Fn On
2		fnfun 5902	. . . . . . 7 ⊢ (ℵ Fn On → Fun ℵ)
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ℵ
4		simpl 472	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5		resfunexg 6384	. . . . . 6 ⊢ ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
6	3, 4, 5	sylancr 694	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7		limelon 5705	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8		onss 6882	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10		fnssres 5918	. . . . . . 7 ⊢ ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
11	1, 9, 10	sylancr 694	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12		fvres 6117	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
13	12	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14		alephord2i 8783	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
15	14	imp 444	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
16	13, 15	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
17	7, 16	sylan 487	. . . . . . . 8 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦 ∈ 𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
18	17	ralrimiva 2949	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19		fnfvrnss 6297	. . . . . . 7 ⊢ (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
20	11, 18, 19	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21		df-f 5808	. . . . . 6 ⊢ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
22	11, 20, 21	sylanbrc 695	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23		alephsmo 8808	. . . . . 6 ⊢ Smo ℵ
24		fndm 5904	. . . . . . . 8 ⊢ (ℵ Fn On → dom ℵ = On)
25	1, 24	ax-mp 5	. . . . . . 7 ⊢ dom ℵ = On
26	7, 25	syl6eleqr 2699	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
27		smores 7336	. . . . . 6 ⊢ ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
28	23, 26, 27	sylancr 694	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
29		alephlim 8773	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
30	29	eleq2d 2673	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
31		eliun 4460	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦))
32		alephon 8775	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
33	32	onelssi 5753	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
34	33	reximi 2994	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
35	31, 34	sylbi 206	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
36	30, 35	syl6bi 242	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
37	36	ralrimiv 2948	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦))
38		feq1 5939	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
39		smoeq 7334	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
40		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (𝑓‘𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
41	40, 12	sylan9eq 2664	. . . . . . . . . . 11 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑓‘𝑦) = (ℵ‘𝑦))
42	41	sseq2d 3596	. . . . . . . . . 10 ⊢ ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
43	42	rexbidva 3031	. . . . . . . . 9 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
44	43	ralbidv 2969	. . . . . . . 8 ⊢ (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (ℵ‘𝑦)))
45	38, 39, 44	3anbi123d 1391	. . . . . . 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))))
46	45	spcegv 3267	. . . . . 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))))
47	46	imp 444	. . . . 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)))
48	6, 22, 28, 37, 47	syl13anc 1320	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦 ∈ 𝐴 𝑥 ⊆ (𝑓‘𝑦)))
49		alephon 8775	. . . . 5 ⊢ (ℵ‘𝐴) ∈ On
50		cfcof 8979	. . . . 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)))
51	49, 7, 50	sylancr 694	. . . 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)))
52	48, 51	mpd 15	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
53	52	expcom 450	. 2 ⊢ (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
54		cf0 8956	. . 3 ⊢ (cf‘∅) = ∅
55		fvprc 6097	. . . 4 ⊢ (¬ 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
56	55	fveq2d 6107	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
57		fvprc 6097	. . 3 ⊢ (¬ 𝐴 ∈ V → (cf‘𝐴) = ∅)
58	54, 56, 57	3eqtr4a 2670	. 2 ⊢ (¬ 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
59	53, 58	pm2.61d1 170	1 ⊢ (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  dom cdm 5038  ran crn 5039   ↾ cres 5040  Oncon0 5640  Lim wlim 5641  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Smo wsmo 7329  ℵcale 8645  cfccf 8646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649  df-cf 8650  df-acn 8651

This theorem is referenced by:  alephom  9286  winafp  9398