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Theorem alephsing 9769
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 9601). 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 9558 . . . . . . 7 ℵ Fn On
2 fnfun 6432 . . . . . . 7 (ℵ Fn On → Fun ℵ)
31, 2ax-mp 5 . . . . . 6 Fun ℵ
4 simpl 486 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5 resfunexg 6982 . . . . . 6 ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
63, 4, 5sylancr 590 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7 limelon 6229 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8 onss 7518 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
97, 8syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10 fnssres 6453 . . . . . . 7 ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
111, 9, 10sylancr 590 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12 fvres 6687 . . . . . . . . . . 11 (𝑦𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
1312adantl 485 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14 alephord2i 9570 . . . . . . . . . . 11 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
1514imp 410 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
1613, 15eqeltrd 2833 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
177, 16sylan 583 . . . . . . . 8 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
1817ralrimiva 3096 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19 fnfvrnss 6888 . . . . . . 7 (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
2011, 18, 19syl2anc 587 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21 df-f 6337 . . . . . 6 ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
2211, 20, 21sylanbrc 586 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23 alephsmo 9595 . . . . . 6 Smo ℵ
241fndmi 6435 . . . . . . 7 dom ℵ = On
257, 24eleqtrrdi 2844 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26 smores 8011 . . . . . 6 ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
2723, 25, 26sylancr 590 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28 alephlim 9560 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
2928eleq2d 2818 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 𝑦𝐴 (ℵ‘𝑦)))
30 eliun 4882 . . . . . . . 8 (𝑥 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦))
31 alephon 9562 . . . . . . . . . 10 (ℵ‘𝑦) ∈ On
3231onelssi 6275 . . . . . . . . 9 (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
3332reximi 3156 . . . . . . . 8 (∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3430, 33sylbi 220 . . . . . . 7 (𝑥 𝑦𝐴 (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3529, 34syl6bi 256 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
3635ralrimiv 3095 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
37 feq1 6479 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38 smoeq 8009 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39 fveq1 6667 . . . . . . . . . . . 12 (𝑓 = (ℵ ↾ 𝐴) → (𝑓𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
4039, 12sylan9eq 2793 . . . . . . . . . . 11 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑓𝑦) = (ℵ‘𝑦))
4140sseq2d 3907 . . . . . . . . . 10 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
4241rexbidva 3205 . . . . . . . . 9 (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4342ralbidv 3109 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4437, 38, 433anbi123d 1437 . . . . . . 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 3499 . . . . . 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 410 . . . . 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 1373 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)))
48 alephon 9562 . . . . 5 (ℵ‘𝐴) ∈ On
49 cfcof 9767 . . . . 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 590 . . . 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 417 . 2 (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53 cf0 9744 . . 3 (cf‘∅) = ∅
54 fvprc 6660 . . . 4 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
5554fveq2d 6672 . . 3 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56 fvprc 6660 . . 3 𝐴 ∈ V → (cf‘𝐴) = ∅)
5753, 55, 563eqtr4a 2799 . 2 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
5852, 57pm2.61d1 183 1 (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2113  wral 3053  wrex 3054  Vcvv 3397  wss 3841  c0 4209   ciun 4878  dom cdm 5519  ran crn 5520  cres 5521  Oncon0 6166  Lim wlim 6167  Fun wfun 6327   Fn wfn 6328  wf 6329  cfv 6333  Smo wsmo 8004  cale 9431  cfccf 9432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-inf2 9170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-isom 6342  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-smo 8005  df-recs 8030  df-rdg 8068  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-oi 9040  df-har 9087  df-card 9434  df-aleph 9435  df-cf 9436  df-acn 9437
This theorem is referenced by:  alephom  10078  winafp  10190
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