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Theorem alephsing 10273
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 10105). 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.
StepHypRef Expression
1 alephfnon 10062 . . . . . . 7 ℵ Fn On
2 fnfun 6649 . . . . . . 7 (ℵ Fn On → Fun ℵ)
31, 2ax-mp 5 . . . . . 6 Fun ℵ
4 simpl 483 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5 resfunexg 7219 . . . . . 6 ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
63, 4, 5sylancr 587 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7 limelon 6428 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8 onss 7774 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
97, 8syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10 fnssres 6673 . . . . . . 7 ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
111, 9, 10sylancr 587 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12 fvres 6910 . . . . . . . . . . 11 (𝑦𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
1312adantl 482 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14 alephord2i 10074 . . . . . . . . . . 11 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
1514imp 407 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
1613, 15eqeltrd 2833 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
177, 16sylan 580 . . . . . . . 8 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
1817ralrimiva 3146 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19 fnfvrnss 7122 . . . . . . 7 (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
2011, 18, 19syl2anc 584 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21 df-f 6547 . . . . . 6 ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
2211, 20, 21sylanbrc 583 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23 alephsmo 10099 . . . . . 6 Smo ℵ
241fndmi 6653 . . . . . . 7 dom ℵ = On
257, 24eleqtrrdi 2844 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26 smores 8354 . . . . . 6 ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
2723, 25, 26sylancr 587 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28 alephlim 10064 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
2928eleq2d 2819 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 𝑦𝐴 (ℵ‘𝑦)))
30 eliun 5001 . . . . . . . 8 (𝑥 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦))
31 alephon 10066 . . . . . . . . . 10 (ℵ‘𝑦) ∈ On
3231onelssi 6479 . . . . . . . . 9 (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
3332reximi 3084 . . . . . . . 8 (∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3430, 33sylbi 216 . . . . . . 7 (𝑥 𝑦𝐴 (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3529, 34syl6bi 252 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
3635ralrimiv 3145 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
37 feq1 6698 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38 smoeq 8352 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39 fveq1 6890 . . . . . . . . . . . 12 (𝑓 = (ℵ ↾ 𝐴) → (𝑓𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
4039, 12sylan9eq 2792 . . . . . . . . . . 11 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑓𝑦) = (ℵ‘𝑦))
4140sseq2d 4014 . . . . . . . . . 10 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
4241rexbidva 3176 . . . . . . . . 9 (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4342ralbidv 3177 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4437, 38, 433anbi123d 1436 . . . . . . 7 (𝑓 = (ℵ ↾ 𝐴) → ((𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)) ↔ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ∧ Smo (ℵ ↾ 𝐴) ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))))
4544spcegv 3587 . . . . . 6 ((ℵ ↾ 𝐴) ∈ V → (((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ∧ Smo (ℵ ↾ 𝐴) ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦))))
4645imp 407 . . . . 5 (((ℵ ↾ 𝐴) ∈ V ∧ ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ∧ Smo (ℵ ↾ 𝐴) ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)))
476, 22, 27, 36, 46syl13anc 1372 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)))
48 alephon 10066 . . . . 5 (ℵ‘𝐴) ∈ On
49 cfcof 10271 . . . . 5 (((ℵ‘𝐴) ∈ On ∧ 𝐴 ∈ On) → (∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
5048, 7, 49sylancr 587 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → (∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
5147, 50mpd 15 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
5251expcom 414 . 2 (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53 cf0 10248 . . 3 (cf‘∅) = ∅
54 fvprc 6883 . . . 4 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
5554fveq2d 6895 . . 3 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56 fvprc 6883 . . 3 𝐴 ∈ V → (cf‘𝐴) = ∅)
5753, 55, 563eqtr4a 2798 . 2 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
5852, 57pm2.61d1 180 1 (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3061  wrex 3070  Vcvv 3474  wss 3948  c0 4322   ciun 4997  dom cdm 5676  ran crn 5677  cres 5678  Oncon0 6364  Lim wlim 6365  Fun wfun 6537   Fn wfn 6538  wf 6539  cfv 6543  Smo wsmo 8347  cale 9933  cfccf 9934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-smo 8348  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937  df-cf 9938  df-acn 9939
This theorem is referenced by:  alephom  10582  winafp  10694
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