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Theorem alephfp 10085
Description: The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 10086 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
alephfplem.1 𝐻 = (rec(ℵ, ω) ↾ ω)
Assertion
Ref Expression
alephfp (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω)

Proof of Theorem alephfp
Dummy variables 𝑧 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfplem.1 . . 3 𝐻 = (rec(ℵ, ω) ↾ ω)
21alephfplem4 10084 . 2 (𝐻 “ ω) ∈ ran ℵ
3 isinfcard 10069 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) ↔ (𝐻 “ ω) ∈ ran ℵ)
4 cardalephex 10067 . . . 4 (ω ⊆ (𝐻 “ ω) → ((card‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧)))
54biimpa 477 . . 3 ((ω ⊆ (𝐻 “ ω) ∧ (card‘ (𝐻 “ ω)) = (𝐻 “ ω)) → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
63, 5sylbir 234 . 2 ( (𝐻 “ ω) ∈ ran ℵ → ∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧))
7 alephle 10065 . . . . . . . . 9 (𝑧 ∈ On → 𝑧 ⊆ (ℵ‘𝑧))
8 alephon 10046 . . . . . . . . . . 11 (ℵ‘𝑧) ∈ On
98onirri 6466 . . . . . . . . . 10 ¬ (ℵ‘𝑧) ∈ (ℵ‘𝑧)
10 frfnom 8417 . . . . . . . . . . . . . 14 (rec(ℵ, ω) ↾ ω) Fn ω
111fneq1i 6635 . . . . . . . . . . . . . 14 (𝐻 Fn ω ↔ (rec(ℵ, ω) ↾ ω) Fn ω)
1210, 11mpbir 230 . . . . . . . . . . . . 13 𝐻 Fn ω
13 fnfun 6638 . . . . . . . . . . . . 13 (𝐻 Fn ω → Fun 𝐻)
14 eluniima 7233 . . . . . . . . . . . . 13 (Fun 𝐻 → (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 (𝐻 “ ω) ↔ ∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣))
16 alephsson 10077 . . . . . . . . . . . . . . . 16 ran ℵ ⊆ On
171alephfplem3 10083 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
1816, 17sselid 3976 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (𝐻𝑣) ∈ On)
19 alephord2i 10054 . . . . . . . . . . . . . . 15 ((𝐻𝑣) ∈ On → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣))))
211alephfplem2 10082 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) = (ℵ‘(𝐻𝑣)))
22 peano2 7863 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
23 fnfvelrn 7067 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn ω ∧ suc 𝑣 ∈ ω) → (𝐻‘suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 688 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ ran 𝐻)
25 fnima 6667 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn ω → (𝐻 “ ω) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 “ ω) = ran 𝐻
2724, 26eleqtrrdi 2843 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ω → (𝐻‘suc 𝑣) ∈ (𝐻 “ ω))
2921, 28eqeltrrd 2833 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω))
30 elssuni 4934 . . . . . . . . . . . . . . . 16 ((ℵ‘(𝐻𝑣)) ∈ (𝐻 “ ω) → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ ω → (ℵ‘(𝐻𝑣)) ⊆ (𝐻 “ ω))
3231sseld 3977 . . . . . . . . . . . . . 14 (𝑣 ∈ ω → ((ℵ‘𝑧) ∈ (ℵ‘(𝐻𝑣)) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3320, 32syld 47 . . . . . . . . . . . . 13 (𝑣 ∈ ω → (𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω)))
3433rexlimiv 3147 . . . . . . . . . . . 12 (∃𝑣 ∈ ω 𝑧 ∈ (𝐻𝑣) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
3515, 34sylbi 216 . . . . . . . . . . 11 (𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω))
36 eleq2 2821 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 (𝐻 “ ω) ↔ 𝑧 ∈ (ℵ‘𝑧)))
37 eleq2 2821 . . . . . . . . . . . 12 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘𝑧) ∈ (𝐻 “ ω) ↔ (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
3836, 37imbi12d 344 . . . . . . . . . . 11 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((𝑧 (𝐻 “ ω) → (ℵ‘𝑧) ∈ (𝐻 “ ω)) ↔ (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧))))
3935, 38mpbii 232 . . . . . . . . . 10 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 ∈ (ℵ‘𝑧) → (ℵ‘𝑧) ∈ (ℵ‘𝑧)))
409, 39mtoi 198 . . . . . . . . 9 ( (𝐻 “ ω) = (ℵ‘𝑧) → ¬ 𝑧 ∈ (ℵ‘𝑧))
417, 40anim12i 613 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧)))
42 eloni 6363 . . . . . . . . . 10 (𝑧 ∈ On → Ord 𝑧)
438onordi 6464 . . . . . . . . . 10 Ord (ℵ‘𝑧)
44 ordtri4 6390 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4542, 43, 44sylancl 586 . . . . . . . . 9 (𝑧 ∈ On → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4645adantr 481 . . . . . . . 8 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (ℵ‘𝑧) ↔ (𝑧 ⊆ (ℵ‘𝑧) ∧ ¬ 𝑧 ∈ (ℵ‘𝑧))))
4741, 46mpbird 256 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (ℵ‘𝑧))
48 eqeq2 2743 . . . . . . . 8 ( (𝐻 “ ω) = (ℵ‘𝑧) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
4948adantl 482 . . . . . . 7 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝑧 = (𝐻 “ ω) ↔ 𝑧 = (ℵ‘𝑧)))
5047, 49mpbird 256 . . . . . 6 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → 𝑧 = (𝐻 “ ω))
5150eqcomd 2737 . . . . 5 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (𝐻 “ ω) = 𝑧)
5251fveq2d 6882 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧))
53 eqeq2 2743 . . . . 5 ( (𝐻 “ ω) = (ℵ‘𝑧) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5453adantl 482 . . . 4 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → ((ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω) ↔ (ℵ‘ (𝐻 “ ω)) = (ℵ‘𝑧)))
5552, 54mpbird 256 . . 3 ((𝑧 ∈ On ∧ (𝐻 “ ω) = (ℵ‘𝑧)) → (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω))
5655rexlimiva 3146 . 2 (∃𝑧 ∈ On (𝐻 “ ω) = (ℵ‘𝑧) → (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω))
572, 6, 56mp2b 10 1 (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3069  wss 3944   cuni 4901  ran crn 5670  cres 5671  cima 5672  Ord word 6352  Oncon0 6353  suc csuc 6355  Fun wfun 6526   Fn wfn 6527  cfv 6532  ωcom 7838  reccrdg 8391  cardccrd 9912  cale 9913
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-oi 9487  df-har 9534  df-card 9916  df-aleph 9917
This theorem is referenced by:  alephfp2  10086
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