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Theorem alephfp 7731
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 7732 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
Dummy variables  z 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfplem.1 . . 3  |-  H  =  ( rec ( aleph ,  om )  |`  om )
21alephfplem4 7730 . 2  |-  U. ( H " om )  e. 
ran  aleph
3 isinfcard 7715 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  <->  U. ( H " om )  e.  ran  aleph )
4 cardalephex 7713 . . . 4  |-  ( om  C_  U. ( H " om )  ->  ( (
card `  U. ( H
" om ) )  =  U. ( H
" om )  <->  E. z  e.  On  U. ( H
" om )  =  ( aleph `  z )
) )
54biimpa 470 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  ->  E. z  e.  On  U. ( H " om )  =  ( aleph `  z ) )
63, 5sylbir 204 . 2  |-  ( U. ( H " om )  e.  ran  aleph  ->  E. z  e.  On  U. ( H
" om )  =  ( aleph `  z )
)
7 alephle 7711 . . . . . . . . 9  |-  ( z  e.  On  ->  z  C_  ( aleph `  z )
)
8 alephon 7692 . . . . . . . . . . 11  |-  ( aleph `  z )  e.  On
98onirri 4498 . . . . . . . . . 10  |-  -.  ( aleph `  z )  e.  ( aleph `  z )
10 frfnom 6443 . . . . . . . . . . . . . 14  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
111fneq1i 5304 . . . . . . . . . . . . . 14  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
1210, 11mpbir 200 . . . . . . . . . . . . 13  |-  H  Fn  om
13 fnfun 5307 . . . . . . . . . . . . 13  |-  ( H  Fn  om  ->  Fun  H )
14 eluniima 5738 . . . . . . . . . . . . 13  |-  ( Fun 
H  ->  ( z  e.  U. ( H " om )  <->  E. v  e.  om  z  e.  ( H `  v ) ) )
1512, 13, 14mp2b 9 . . . . . . . . . . . 12  |-  ( z  e.  U. ( H
" om )  <->  E. v  e.  om  z  e.  ( H `  v ) )
16 alephsson 7723 . . . . . . . . . . . . . . . 16  |-  ran  aleph  C_  On
171alephfplem3 7729 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
1816, 17sseldi 3179 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( H `  v )  e.  On )
19 alephord2i 7700 . . . . . . . . . . . . . . 15  |-  ( ( H `  v )  e.  On  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  ( aleph `  ( H `  v ) ) ) )
2018, 19syl 15 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  ( aleph `  ( H `  v ) ) ) )
211alephfplem2 7728 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  =  ( aleph `  ( H `  v )
) )
22 peano2 4675 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  om  ->  suc  v  e.  om )
23 fnfvelrn 5624 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  Fn  om  /\  suc  v  e.  om )  ->  ( H `  suc  v )  e.  ran  H )
2412, 23mpan 651 . . . . . . . . . . . . . . . . . . 19  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ran  H
)
25 fnima 5328 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  Fn  om  ->  ( H " om )  =  ran  H )
2612, 25ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( H
" om )  =  ran  H
2724, 26syl6eleqr 2375 . . . . . . . . . . . . . . . . . 18  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ( H
" om ) )
2822, 27syl 15 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  e.  ( H " om ) )
2921, 28eqeltrrd 2359 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  e.  ( H " om ) )
30 elssuni 3856 . . . . . . . . . . . . . . . 16  |-  ( (
aleph `  ( H `  v ) )  e.  ( H " om )  ->  ( aleph `  ( H `  v )
)  C_  U. ( H " om ) )
3129, 30syl 15 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  C_  U. ( H " om ) )
3231sseld 3180 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
( aleph `  z )  e.  ( aleph `  ( H `  v ) )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3320, 32syld 40 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3433rexlimiv 2662 . . . . . . . . . . . 12  |-  ( E. v  e.  om  z  e.  ( H `  v
)  ->  ( aleph `  z )  e.  U. ( H " om )
)
3515, 34sylbi 187 . . . . . . . . . . 11  |-  ( z  e.  U. ( H
" om )  -> 
( aleph `  z )  e.  U. ( H " om ) )
36 eleq2 2345 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  U. ( H " om )  <->  z  e.  (
aleph `  z ) ) )
37 eleq2 2345 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( ( aleph `  z )  e. 
U. ( H " om )  <->  ( aleph `  z
)  e.  ( aleph `  z ) ) )
3836, 37imbi12d 311 . . . . . . . . . . 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 202 . . . . . . . . . 10  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  ( aleph `  z )  ->  ( aleph `  z )  e.  ( aleph `  z )
) )
409, 39mtoi 169 . . . . . . . . 9  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  -.  z  e.  ( aleph `  z )
)
417, 40anim12i 549 . . . . . . . 8  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  C_  ( aleph `  z )  /\  -.  z  e.  ( aleph `  z ) ) )
42 eloni 4401 . . . . . . . . . 10  |-  ( z  e.  On  ->  Ord  z )
438onordi 4496 . . . . . . . . . 10  |-  Ord  ( aleph `  z )
44 ordtri4 4428 . . . . . . . . . 10  |-  ( ( Ord  z  /\  Ord  ( aleph `  z )
)  ->  ( z  =  ( aleph `  z
)  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4542, 43, 44sylancl 643 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  =  ( aleph `  z )  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4645adantr 451 . . . . . . . 8  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  =  (
aleph `  z )  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4741, 46mpbird 223 . . . . . . 7  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
z  =  ( aleph `  z ) )
48 eqeq2 2293 . . . . . . . 8  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  =  U. ( H " om )  <->  z  =  (
aleph `  z ) ) )
4948adantl 452 . . . . . . 7  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  =  U. ( H " om )  <->  z  =  ( aleph `  z
) ) )
5047, 49mpbird 223 . . . . . 6  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
z  =  U. ( H " om ) )
5150eqcomd 2289 . . . . 5  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  ->  U. ( H " om )  =  z )
5251fveq2d 5490 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  ( aleph `  z
) )
53 eqeq2 2293 . . . . 5  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( ( aleph `  U. ( H
" om ) )  =  U. ( H
" om )  <->  ( aleph ` 
U. ( H " om ) )  =  (
aleph `  z ) ) )
5453adantl 452 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( ( aleph `  U. ( H " om )
)  =  U. ( H " om )  <->  ( aleph ` 
U. ( H " om ) )  =  (
aleph `  z ) ) )
5552, 54mpbird 223 . . 3  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  U. ( H
" om ) )
5655rexlimiva 2663 . 2  |-  ( E. z  e.  On  U. ( H " om )  =  ( aleph `  z
)  ->  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )
)
572, 6, 56mp2b 9 1  |-  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   E.wrex 2545    C_ wss 3153   U.cuni 3828   Ord word 4390   Oncon0 4391   suc csuc 4393   omcom 4655   ran crn 4689    |` cres 4690   "cima 4691   Fun wfun 5215    Fn wfn 5216   ` cfv 5221   reccrdg 6418   cardccrd 7564   alephcale 7565
This theorem is referenced by:  alephfp2  7732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-har 7268  df-card 7568  df-aleph 7569
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