MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephsing Structured version   Visualization version   GIF version

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.
StepHypRef Expression
1 alephfnon 10057 . . . . . . 7 ℵ Fn On
2 fnfun 6640 . . . . . . 7 (ℵ Fn On → Fun ℵ)
31, 2ax-mp 5 . . . . . 6 Fun ℵ
4 simpl 482 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5 resfunexg 7209 . . . . . 6 ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
63, 4, 5sylancr 586 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7 limelon 6419 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8 onss 7766 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
97, 8syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10 fnssres 6664 . . . . . . 7 ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
111, 9, 10sylancr 586 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12 fvres 6901 . . . . . . . . . . 11 (𝑦𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
1312adantl 481 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14 alephord2i 10069 . . . . . . . . . . 11 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
1514imp 406 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
1613, 15eqeltrd 2825 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
177, 16sylan 579 . . . . . . . 8 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
1817ralrimiva 3138 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19 fnfvrnss 7113 . . . . . . 7 (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
2011, 18, 19syl2anc 583 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21 df-f 6538 . . . . . 6 ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
2211, 20, 21sylanbrc 582 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23 alephsmo 10094 . . . . . 6 Smo ℵ
241fndmi 6644 . . . . . . 7 dom ℵ = On
257, 24eleqtrrdi 2836 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26 smores 8348 . . . . . 6 ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
2723, 25, 26sylancr 586 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28 alephlim 10059 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
2928eleq2d 2811 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 𝑦𝐴 (ℵ‘𝑦)))
30 eliun 4992 . . . . . . . 8 (𝑥 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦))
31 alephon 10061 . . . . . . . . . 10 (ℵ‘𝑦) ∈ On
3231onelssi 6470 . . . . . . . . 9 (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
3332reximi 3076 . . . . . . . 8 (∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3430, 33sylbi 216 . . . . . . 7 (𝑥 𝑦𝐴 (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3529, 34biimtrdi 252 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
3635ralrimiv 3137 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
37 feq1 6689 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38 smoeq 8346 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39 fveq1 6881 . . . . . . . . . . . 12 (𝑓 = (ℵ ↾ 𝐴) → (𝑓𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
4039, 12sylan9eq 2784 . . . . . . . . . . 11 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑓𝑦) = (ℵ‘𝑦))
4140sseq2d 4007 . . . . . . . . . 10 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
4241rexbidva 3168 . . . . . . . . 9 (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4342ralbidv 3169 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4437, 38, 433anbi123d 1432 . . . . . . 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 3579 . . . . . 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 406 . . . . 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 1369 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)))
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)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
5048, 7, 49sylancr 586 . . . 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 413 . 2 (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53 cf0 10243 . . 3 (cf‘∅) = ∅
54 fvprc 6874 . . . 4 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
5554fveq2d 6886 . . 3 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56 fvprc 6874 . . 3 𝐴 ∈ V → (cf‘𝐴) = ∅)
5753, 55, 563eqtr4a 2790 . 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 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wral 3053  wrex 3062  Vcvv 3466  wss 3941  c0 4315   ciun 4988  dom cdm 5667  ran crn 5668  cres 5669  Oncon0 6355  Lim wlim 6356  Fun wfun 6528   Fn wfn 6529  wf 6530  cfv 6534  Smo wsmo 8341  cale 9928  cfccf 9929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-smo 8342  df-recs 8367  df-rdg 8406  df-1o 8462  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
  Copyright terms: Public domain W3C validator