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Theorem alephsing 10313
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 10145). 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 10102 . . . . . . 7 ℵ Fn On
2 fnfun 6668 . . . . . . 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 7234 . . . . . 6 ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
63, 4, 5sylancr 587 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7 limelon 6449 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8 onss 7803 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
97, 8syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10 fnssres 6691 . . . . . . 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 6925 . . . . . . . . . . 11 (𝑦𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
1312adantl 481 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14 alephord2i 10114 . . . . . . . . . . 11 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
1514imp 406 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
1613, 15eqeltrd 2838 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
177, 16sylan 580 . . . . . . . 8 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
1817ralrimiva 3143 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19 fnfvrnss 7140 . . . . . . 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 6566 . . . . . 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 10139 . . . . . 6 Smo ℵ
241fndmi 6672 . . . . . . 7 dom ℵ = On
257, 24eleqtrrdi 2849 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
26 smores 8390 . . . . . 6 ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
2723, 25, 26sylancr 587 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
28 alephlim 10104 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
2928eleq2d 2824 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 𝑦𝐴 (ℵ‘𝑦)))
30 eliun 4999 . . . . . . . 8 (𝑥 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦))
31 alephon 10106 . . . . . . . . . 10 (ℵ‘𝑦) ∈ On
3231onelssi 6500 . . . . . . . . 9 (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
3332reximi 3081 . . . . . . . 8 (∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3430, 33sylbi 217 . . . . . . 7 (𝑥 𝑦𝐴 (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3529, 34biimtrdi 253 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
3635ralrimiv 3142 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
37 feq1 6716 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
38 smoeq 8388 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
39 fveq1 6905 . . . . . . . . . . . 12 (𝑓 = (ℵ ↾ 𝐴) → (𝑓𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
4039, 12sylan9eq 2794 . . . . . . . . . . 11 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑓𝑦) = (ℵ‘𝑦))
4140sseq2d 4027 . . . . . . . . . 10 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
4241rexbidva 3174 . . . . . . . . 9 (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4342ralbidv 3175 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4437, 38, 433anbi123d 1435 . . . . . . 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 3596 . . . . . 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 1371 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)))
48 alephon 10106 . . . . 5 (ℵ‘𝐴) ∈ On
49 cfcof 10311 . . . . 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 413 . 2 (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
53 cf0 10288 . . 3 (cf‘∅) = ∅
54 fvprc 6898 . . . 4 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
5554fveq2d 6910 . . 3 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
56 fvprc 6898 . . 3 𝐴 ∈ V → (cf‘𝐴) = ∅)
5753, 55, 563eqtr4a 2800 . 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 1086   = wceq 1536  wex 1775  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  wss 3962  c0 4338   ciun 4995  dom cdm 5688  ran crn 5689  cres 5690  Oncon0 6385  Lim wlim 6386  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  Smo wsmo 8383  cale 9973  cfccf 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-smo 8384  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-oi 9547  df-har 9594  df-card 9976  df-aleph 9977  df-cf 9978  df-acn 9979
This theorem is referenced by:  alephom  10622  winafp  10734
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