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Theorem alephfp 10099
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 10100 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(β„΅, Ο‰) β†Ύ Ο‰)
Assertion
Ref Expression
alephfp (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)
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(β„΅, Ο‰) β†Ύ Ο‰)
21alephfplem4 10098 . 2 βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅
3 isinfcard 10083 . . 3 ((Ο‰ βŠ† βˆͺ (𝐻 β€œ Ο‰) ∧ (cardβ€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)) ↔ βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅)
4 cardalephex 10081 . . . 4 (Ο‰ βŠ† βˆͺ (𝐻 β€œ Ο‰) β†’ ((cardβ€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰) ↔ βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)))
54biimpa 478 . . 3 ((Ο‰ βŠ† βˆͺ (𝐻 β€œ Ο‰) ∧ (cardβ€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)) β†’ βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§))
63, 5sylbir 234 . 2 (βˆͺ (𝐻 β€œ Ο‰) ∈ ran β„΅ β†’ βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§))
7 alephle 10079 . . . . . . . . 9 (𝑧 ∈ On β†’ 𝑧 βŠ† (β„΅β€˜π‘§))
8 alephon 10060 . . . . . . . . . . 11 (β„΅β€˜π‘§) ∈ On
98onirri 6474 . . . . . . . . . 10 Β¬ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)
10 frfnom 8430 . . . . . . . . . . . . . 14 (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰
111fneq1i 6643 . . . . . . . . . . . . . 14 (𝐻 Fn Ο‰ ↔ (rec(β„΅, Ο‰) β†Ύ Ο‰) Fn Ο‰)
1210, 11mpbir 230 . . . . . . . . . . . . 13 𝐻 Fn Ο‰
13 fnfun 6646 . . . . . . . . . . . . 13 (𝐻 Fn Ο‰ β†’ Fun 𝐻)
14 eluniima 7244 . . . . . . . . . . . . 13 (Fun 𝐻 β†’ (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ βˆƒπ‘£ ∈ Ο‰ 𝑧 ∈ (π»β€˜π‘£)))
1512, 13, 14mp2b 10 . . . . . . . . . . . 12 (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ βˆƒπ‘£ ∈ Ο‰ 𝑧 ∈ (π»β€˜π‘£))
16 alephsson 10091 . . . . . . . . . . . . . . . 16 ran β„΅ βŠ† On
171alephfplem3 10097 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ ran β„΅)
1816, 17sselid 3979 . . . . . . . . . . . . . . 15 (𝑣 ∈ Ο‰ β†’ (π»β€˜π‘£) ∈ On)
19 alephord2i 10068 . . . . . . . . . . . . . . 15 ((π»β€˜π‘£) ∈ On β†’ (𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜(π»β€˜π‘£))))
2018, 19syl 17 . . . . . . . . . . . . . 14 (𝑣 ∈ Ο‰ β†’ (𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜(π»β€˜π‘£))))
211alephfplem2 10096 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) = (β„΅β€˜(π»β€˜π‘£)))
22 peano2 7876 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ Ο‰ β†’ suc 𝑣 ∈ Ο‰)
23 fnfvelrn 7078 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 Fn Ο‰ ∧ suc 𝑣 ∈ Ο‰) β†’ (π»β€˜suc 𝑣) ∈ ran 𝐻)
2412, 23mpan 689 . . . . . . . . . . . . . . . . . . 19 (suc 𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ ran 𝐻)
25 fnima 6677 . . . . . . . . . . . . . . . . . . . 20 (𝐻 Fn Ο‰ β†’ (𝐻 β€œ Ο‰) = ran 𝐻)
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (𝐻 β€œ Ο‰) = ran 𝐻
2724, 26eleqtrrdi 2845 . . . . . . . . . . . . . . . . . 18 (suc 𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ (𝐻 β€œ Ο‰))
2822, 27syl 17 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ Ο‰ β†’ (π»β€˜suc 𝑣) ∈ (𝐻 β€œ Ο‰))
2921, 28eqeltrrd 2835 . . . . . . . . . . . . . . . 16 (𝑣 ∈ Ο‰ β†’ (β„΅β€˜(π»β€˜π‘£)) ∈ (𝐻 β€œ Ο‰))
30 elssuni 4940 . . . . . . . . . . . . . . . 16 ((β„΅β€˜(π»β€˜π‘£)) ∈ (𝐻 β€œ Ο‰) β†’ (β„΅β€˜(π»β€˜π‘£)) βŠ† βˆͺ (𝐻 β€œ Ο‰))
3129, 30syl 17 . . . . . . . . . . . . . . 15 (𝑣 ∈ Ο‰ β†’ (β„΅β€˜(π»β€˜π‘£)) βŠ† βˆͺ (𝐻 β€œ Ο‰))
3231sseld 3980 . . . . . . . . . . . . . 14 (𝑣 ∈ Ο‰ β†’ ((β„΅β€˜π‘§) ∈ (β„΅β€˜(π»β€˜π‘£)) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰)))
3320, 32syld 47 . . . . . . . . . . . . 13 (𝑣 ∈ Ο‰ β†’ (𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰)))
3433rexlimiv 3149 . . . . . . . . . . . 12 (βˆƒπ‘£ ∈ Ο‰ 𝑧 ∈ (π»β€˜π‘£) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰))
3515, 34sylbi 216 . . . . . . . . . . 11 (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰))
36 eleq2 2823 . . . . . . . . . . . 12 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 ∈ (β„΅β€˜π‘§)))
37 eleq2 2823 . . . . . . . . . . . 12 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ ((β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰) ↔ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)))
3836, 37imbi12d 345 . . . . . . . . . . 11 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ ((𝑧 ∈ βˆͺ (𝐻 β€œ Ο‰) β†’ (β„΅β€˜π‘§) ∈ βˆͺ (𝐻 β€œ Ο‰)) ↔ (𝑧 ∈ (β„΅β€˜π‘§) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§))))
3935, 38mpbii 232 . . . . . . . . . 10 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 ∈ (β„΅β€˜π‘§) β†’ (β„΅β€˜π‘§) ∈ (β„΅β€˜π‘§)))
409, 39mtoi 198 . . . . . . . . 9 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))
417, 40anim12i 614 . . . . . . . 8 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§)))
42 eloni 6371 . . . . . . . . . 10 (𝑧 ∈ On β†’ Ord 𝑧)
438onordi 6472 . . . . . . . . . 10 Ord (β„΅β€˜π‘§)
44 ordtri4 6398 . . . . . . . . . 10 ((Ord 𝑧 ∧ Ord (β„΅β€˜π‘§)) β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4542, 43, 44sylancl 587 . . . . . . . . 9 (𝑧 ∈ On β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4645adantr 482 . . . . . . . 8 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 = (β„΅β€˜π‘§) ↔ (𝑧 βŠ† (β„΅β€˜π‘§) ∧ Β¬ 𝑧 ∈ (β„΅β€˜π‘§))))
4741, 46mpbird 257 . . . . . . 7 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ 𝑧 = (β„΅β€˜π‘§))
48 eqeq2 2745 . . . . . . . 8 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
4948adantl 483 . . . . . . 7 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (𝑧 = βˆͺ (𝐻 β€œ Ο‰) ↔ 𝑧 = (β„΅β€˜π‘§)))
5047, 49mpbird 257 . . . . . 6 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ 𝑧 = βˆͺ (𝐻 β€œ Ο‰))
5150eqcomd 2739 . . . . 5 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ βˆͺ (𝐻 β€œ Ο‰) = 𝑧)
5251fveq2d 6892 . . . 4 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§))
53 eqeq2 2745 . . . . 5 (βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ ((β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰) ↔ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§)))
5453adantl 483 . . . 4 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ ((β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰) ↔ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = (β„΅β€˜π‘§)))
5552, 54mpbird 257 . . 3 ((𝑧 ∈ On ∧ βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§)) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰))
5655rexlimiva 3148 . 2 (βˆƒπ‘§ ∈ On βˆͺ (𝐻 β€œ Ο‰) = (β„΅β€˜π‘§) β†’ (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰))
572, 6, 56mp2b 10 1 (β„΅β€˜βˆͺ (𝐻 β€œ Ο‰)) = βˆͺ (𝐻 β€œ Ο‰)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βŠ† wss 3947  βˆͺ cuni 4907  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  Ord word 6360  Oncon0 6361  suc csuc 6363  Fun wfun 6534   Fn wfn 6535  β€˜cfv 6540  Ο‰com 7850  reccrdg 8404  cardccrd 9926  β„΅cale 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-oi 9501  df-har 9548  df-card 9930  df-aleph 9931
This theorem is referenced by:  alephfp2  10100
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