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Theorem alephfp 7668
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 7669 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 ( aleph ,  om )  |`  om )
Assertion
Ref Expression
alephfp  |-  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )

Proof of Theorem alephfp
StepHypRef Expression
1 alephfplem.1 . . 3  |-  H  =  ( rec ( aleph ,  om )  |`  om )
21alephfplem4 7667 . 2  |-  U. ( H " om )  e. 
ran  aleph
3 isinfcard 7652 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  <->  U. ( H " om )  e.  ran  aleph )
4 cardalephex 7650 . . . 4  |-  ( om  C_  U. ( H " om )  ->  ( (
card `  U. ( H
" om ) )  =  U. ( H
" om )  <->  E. z  e.  On  U. ( H
" om )  =  ( aleph `  z )
) )
54biimpa 472 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  ->  E. z  e.  On  U. ( H " om )  =  ( aleph `  z ) )
63, 5sylbir 206 . 2  |-  ( U. ( H " om )  e.  ran  aleph  ->  E. z  e.  On  U. ( H
" om )  =  ( aleph `  z )
)
7 alephle 7648 . . . . . . . . 9  |-  ( z  e.  On  ->  z  C_  ( aleph `  z )
)
8 alephon 7629 . . . . . . . . . . 11  |-  ( aleph `  z )  e.  On
98onirri 4436 . . . . . . . . . 10  |-  -.  ( aleph `  z )  e.  ( aleph `  z )
10 frfnom 6380 . . . . . . . . . . . . . 14  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
111fneq1i 5241 . . . . . . . . . . . . . 14  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
1210, 11mpbir 202 . . . . . . . . . . . . 13  |-  H  Fn  om
13 fnfun 5244 . . . . . . . . . . . . 13  |-  ( H  Fn  om  ->  Fun  H )
14 eluniima 5675 . . . . . . . . . . . . 13  |-  ( Fun 
H  ->  ( z  e.  U. ( H " om )  <->  E. v  e.  om  z  e.  ( H `  v ) ) )
1512, 13, 14mp2b 11 . . . . . . . . . . . 12  |-  ( z  e.  U. ( H
" om )  <->  E. v  e.  om  z  e.  ( H `  v ) )
16 alephsson 7660 . . . . . . . . . . . . . . . 16  |-  ran  aleph  C_  On
171alephfplem3 7666 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
1816, 17sseldi 3120 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( H `  v )  e.  On )
19 alephord2i 7637 . . . . . . . . . . . . . . 15  |-  ( ( H `  v )  e.  On  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  ( aleph `  ( H `  v ) ) ) )
2018, 19syl 17 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  ( aleph `  ( H `  v ) ) ) )
211alephfplem2 7665 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  =  ( aleph `  ( H `  v )
) )
22 peano2 4613 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  om  ->  suc  v  e.  om )
23 fnfvelrn 5561 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  Fn  om  /\  suc  v  e.  om )  ->  ( H `  suc  v )  e.  ran  H )
2412, 23mpan 654 . . . . . . . . . . . . . . . . . . 19  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ran  H
)
25 fnima 5265 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  Fn  om  ->  ( H " om )  =  ran  H )
2612, 25ax-mp 10 . . . . . . . . . . . . . . . . . . 19  |-  ( H
" om )  =  ran  H
2724, 26syl6eleqr 2347 . . . . . . . . . . . . . . . . . 18  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ( H
" om ) )
2822, 27syl 17 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  e.  ( H " om ) )
2921, 28eqeltrrd 2331 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  e.  ( H " om ) )
30 elssuni 3796 . . . . . . . . . . . . . . . 16  |-  ( (
aleph `  ( H `  v ) )  e.  ( H " om )  ->  ( aleph `  ( H `  v )
)  C_  U. ( H " om ) )
3129, 30syl 17 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  C_  U. ( H " om ) )
3231sseld 3121 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
( aleph `  z )  e.  ( aleph `  ( H `  v ) )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3320, 32syld 42 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3433rexlimiv 2632 . . . . . . . . . . . 12  |-  ( E. v  e.  om  z  e.  ( H `  v
)  ->  ( aleph `  z )  e.  U. ( H " om )
)
3515, 34sylbi 189 . . . . . . . . . . 11  |-  ( z  e.  U. ( H
" om )  -> 
( aleph `  z )  e.  U. ( H " om ) )
36 eleq2 2317 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  U. ( H " om )  <->  z  e.  (
aleph `  z ) ) )
37 eleq2 2317 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( ( aleph `  z )  e. 
U. ( H " om )  <->  ( aleph `  z
)  e.  ( aleph `  z ) ) )
3836, 37imbi12d 313 . . . . . . . . . . 11  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( (
z  e.  U. ( H " om )  -> 
( aleph `  z )  e.  U. ( H " om ) )  <->  ( z  e.  ( aleph `  z )  ->  ( aleph `  z )  e.  ( aleph `  z )
) ) )
3935, 38mpbii 204 . . . . . . . . . 10  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  ( aleph `  z )  ->  ( aleph `  z )  e.  ( aleph `  z )
) )
409, 39mtoi 171 . . . . . . . . 9  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  -.  z  e.  ( aleph `  z )
)
417, 40anim12i 551 . . . . . . . 8  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  C_  ( aleph `  z )  /\  -.  z  e.  ( aleph `  z ) ) )
42 eloni 4339 . . . . . . . . . 10  |-  ( z  e.  On  ->  Ord  z )
438onordi 4434 . . . . . . . . . 10  |-  Ord  ( aleph `  z )
44 ordtri4 4366 . . . . . . . . . 10  |-  ( ( Ord  z  /\  Ord  ( aleph `  z )
)  ->  ( z  =  ( aleph `  z
)  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4542, 43, 44sylancl 646 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  =  ( aleph `  z )  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4645adantr 453 . . . . . . . 8  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  =  (
aleph `  z )  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4741, 46mpbird 225 . . . . . . 7  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
z  =  ( aleph `  z ) )
48 eqeq2 2265 . . . . . . . 8  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  =  U. ( H " om )  <->  z  =  (
aleph `  z ) ) )
4948adantl 454 . . . . . . 7  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  =  U. ( H " om )  <->  z  =  ( aleph `  z
) ) )
5047, 49mpbird 225 . . . . . 6  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
z  =  U. ( H " om ) )
5150eqcomd 2261 . . . . 5  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  ->  U. ( H " om )  =  z )
5251fveq2d 5427 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  ( aleph `  z
) )
53 eqeq2 2265 . . . . 5  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( ( aleph `  U. ( H
" om ) )  =  U. ( H
" om )  <->  ( aleph ` 
U. ( H " om ) )  =  (
aleph `  z ) ) )
5453adantl 454 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( ( aleph `  U. ( H " om )
)  =  U. ( H " om )  <->  ( aleph ` 
U. ( H " om ) )  =  (
aleph `  z ) ) )
5552, 54mpbird 225 . . 3  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  U. ( H
" om ) )
5655rexlimiva 2633 . 2  |-  ( E. z  e.  On  U. ( H " om )  =  ( aleph `  z
)  ->  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )
)
572, 6, 56mp2b 11 1  |-  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517    C_ wss 3094   U.cuni 3768   Ord word 4328   Oncon0 4329   suc csuc 4331   omcom 4593   ran crn 4627    |` cres 4628   "cima 4629   Fun wfun 4632    Fn wfn 4633   ` cfv 4638   reccrdg 6355   cardccrd 7501   alephcale 7502
This theorem is referenced by:  alephfp2  7669
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-har 7205  df-card 7505  df-aleph 7506
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