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Theorem alephsing 9050
 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 8883). 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 8840 . . . . . . 7 ℵ Fn On
2 fnfun 5951 . . . . . . 7 (ℵ Fn On → Fun ℵ)
31, 2ax-mp 5 . . . . . 6 Fun ℵ
4 simpl 473 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ V)
5 resfunexg 6439 . . . . . 6 ((Fun ℵ ∧ 𝐴 ∈ V) → (ℵ ↾ 𝐴) ∈ V)
63, 4, 5sylancr 694 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) ∈ V)
7 limelon 5752 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
8 onss 6944 . . . . . . . 8 (𝐴 ∈ On → 𝐴 ⊆ On)
97, 8syl 17 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ⊆ On)
10 fnssres 5967 . . . . . . 7 ((ℵ Fn On ∧ 𝐴 ⊆ On) → (ℵ ↾ 𝐴) Fn 𝐴)
111, 9, 10sylancr 694 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴) Fn 𝐴)
12 fvres 6169 . . . . . . . . . . 11 (𝑦𝐴 → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
1312adantl 482 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) = (ℵ‘𝑦))
14 alephord2i 8852 . . . . . . . . . . 11 (𝐴 ∈ On → (𝑦𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
1514imp 445 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
1613, 15eqeltrd 2698 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
177, 16sylan 488 . . . . . . . 8 (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑦𝐴) → ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
1817ralrimiva 2961 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴))
19 fnfvrnss 6351 . . . . . . 7 (((ℵ ↾ 𝐴) Fn 𝐴 ∧ ∀𝑦𝐴 ((ℵ ↾ 𝐴)‘𝑦) ∈ (ℵ‘𝐴)) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
2011, 18, 19syl2anc 692 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴))
21 df-f 5856 . . . . . 6 ((ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴) ↔ ((ℵ ↾ 𝐴) Fn 𝐴 ∧ ran (ℵ ↾ 𝐴) ⊆ (ℵ‘𝐴)))
2211, 20, 21sylanbrc 697 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴))
23 alephsmo 8877 . . . . . 6 Smo ℵ
24 fndm 5953 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
251, 24ax-mp 5 . . . . . . 7 dom ℵ = On
267, 25syl6eleqr 2709 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ dom ℵ)
27 smores 7401 . . . . . 6 ((Smo ℵ ∧ 𝐴 ∈ dom ℵ) → Smo (ℵ ↾ 𝐴))
2823, 26, 27sylancr 694 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → Smo (ℵ ↾ 𝐴))
29 alephlim 8842 . . . . . . . 8 ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑦𝐴 (ℵ‘𝑦))
3029eleq2d 2684 . . . . . . 7 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) ↔ 𝑥 𝑦𝐴 (ℵ‘𝑦)))
31 eliun 4495 . . . . . . . 8 (𝑥 𝑦𝐴 (ℵ‘𝑦) ↔ ∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦))
32 alephon 8844 . . . . . . . . . 10 (ℵ‘𝑦) ∈ On
3332onelssi 5800 . . . . . . . . 9 (𝑥 ∈ (ℵ‘𝑦) → 𝑥 ⊆ (ℵ‘𝑦))
3433reximi 3006 . . . . . . . 8 (∃𝑦𝐴 𝑥 ∈ (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3531, 34sylbi 207 . . . . . . 7 (𝑥 𝑦𝐴 (ℵ‘𝑦) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
3630, 35syl6bi 243 . . . . . 6 ((𝐴 ∈ V ∧ Lim 𝐴) → (𝑥 ∈ (ℵ‘𝐴) → ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
3736ralrimiv 2960 . . . . 5 ((𝐴 ∈ V ∧ Lim 𝐴) → ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦))
38 feq1 5988 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (𝑓:𝐴⟶(ℵ‘𝐴) ↔ (ℵ ↾ 𝐴):𝐴⟶(ℵ‘𝐴)))
39 smoeq 7399 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (Smo 𝑓 ↔ Smo (ℵ ↾ 𝐴)))
40 fveq1 6152 . . . . . . . . . . . 12 (𝑓 = (ℵ ↾ 𝐴) → (𝑓𝑦) = ((ℵ ↾ 𝐴)‘𝑦))
4140, 12sylan9eq 2675 . . . . . . . . . . 11 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑓𝑦) = (ℵ‘𝑦))
4241sseq2d 3617 . . . . . . . . . 10 ((𝑓 = (ℵ ↾ 𝐴) ∧ 𝑦𝐴) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ⊆ (ℵ‘𝑦)))
4342rexbidva 3043 . . . . . . . . 9 (𝑓 = (ℵ ↾ 𝐴) → (∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4443ralbidv 2981 . . . . . . . 8 (𝑓 = (ℵ ↾ 𝐴) → (∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (ℵ‘𝑦)))
4538, 39, 443anbi123d 1396 . . . . . . 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))))
4645spcegv 3283 . . . . . 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))))
4746imp 445 . . . . 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)))
486, 22, 28, 37, 47syl13anc 1325 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → ∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)))
49 alephon 8844 . . . . 5 (ℵ‘𝐴) ∈ On
50 cfcof 9048 . . . . 5 (((ℵ‘𝐴) ∈ On ∧ 𝐴 ∈ On) → (∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
5149, 7, 50sylancr 694 . . . 4 ((𝐴 ∈ V ∧ Lim 𝐴) → (∃𝑓(𝑓:𝐴⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑥 ∈ (ℵ‘𝐴)∃𝑦𝐴 𝑥 ⊆ (𝑓𝑦)) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
5248, 51mpd 15 . . 3 ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
5352expcom 451 . 2 (Lim 𝐴 → (𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴)))
54 cf0 9025 . . 3 (cf‘∅) = ∅
55 fvprc 6147 . . . 4 𝐴 ∈ V → (ℵ‘𝐴) = ∅)
5655fveq2d 6157 . . 3 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘∅))
57 fvprc 6147 . . 3 𝐴 ∈ V → (cf‘𝐴) = ∅)
5854, 56, 573eqtr4a 2681 . 2 𝐴 ∈ V → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
5953, 58pm2.61d1 171 1 (Lim 𝐴 → (cf‘(ℵ‘𝐴)) = (cf‘𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3189   ⊆ wss 3559  ∅c0 3896  ∪ ciun 4490  dom cdm 5079  ran crn 5080   ↾ cres 5081  Oncon0 5687  Lim wlim 5688  Fun wfun 5846   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  Smo wsmo 7394  ℵcale 8714  cfccf 8715 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-smo 7395  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-oi 8367  df-har 8415  df-card 8717  df-aleph 8718  df-cf 8719  df-acn 8720 This theorem is referenced by:  alephom  9359  winafp  9471
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