MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephfp Structured version   Unicode version

Theorem alephfp 7991
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 7992 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 7990 . 2  |-  U. ( H " om )  e. 
ran  aleph
3 isinfcard 7975 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  <->  U. ( H " om )  e.  ran  aleph )
4 cardalephex 7973 . . . 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 7971 . . . . . . . . 9  |-  ( z  e.  On  ->  z  C_  ( aleph `  z )
)
8 alephon 7952 . . . . . . . . . . 11  |-  ( aleph `  z )  e.  On
98onirri 4690 . . . . . . . . . 10  |-  -.  ( aleph `  z )  e.  ( aleph `  z )
10 frfnom 6694 . . . . . . . . . . . . . 14  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
111fneq1i 5541 . . . . . . . . . . . . . 14  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
1210, 11mpbir 202 . . . . . . . . . . . . 13  |-  H  Fn  om
13 fnfun 5544 . . . . . . . . . . . . 13  |-  ( H  Fn  om  ->  Fun  H )
14 eluniima 5999 . . . . . . . . . . . . 13  |-  ( Fun 
H  ->  ( z  e.  U. ( H " om )  <->  E. v  e.  om  z  e.  ( H `  v ) ) )
1512, 13, 14mp2b 10 . . . . . . . . . . . 12  |-  ( z  e.  U. ( H
" om )  <->  E. v  e.  om  z  e.  ( H `  v ) )
16 alephsson 7983 . . . . . . . . . . . . . . . 16  |-  ran  aleph  C_  On
171alephfplem3 7989 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
1816, 17sseldi 3348 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( H `  v )  e.  On )
19 alephord2i 7960 . . . . . . . . . . . . . . 15  |-  ( ( H `  v )  e.  On  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  ( aleph `  ( H `  v ) ) ) )
2018, 19syl 16 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  ( aleph `  ( H `  v ) ) ) )
211alephfplem2 7988 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  =  ( aleph `  ( H `  v )
) )
22 peano2 4867 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  om  ->  suc  v  e.  om )
23 fnfvelrn 5869 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  Fn  om  /\  suc  v  e.  om )  ->  ( H `  suc  v )  e.  ran  H )
2412, 23mpan 653 . . . . . . . . . . . . . . . . . . 19  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ran  H
)
25 fnima 5565 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  Fn  om  ->  ( H " om )  =  ran  H )
2612, 25ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( H
" om )  =  ran  H
2724, 26syl6eleqr 2529 . . . . . . . . . . . . . . . . . 18  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ( H
" om ) )
2822, 27syl 16 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  e.  ( H " om ) )
2921, 28eqeltrrd 2513 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  e.  ( H " om ) )
30 elssuni 4045 . . . . . . . . . . . . . . . 16  |-  ( (
aleph `  ( H `  v ) )  e.  ( H " om )  ->  ( aleph `  ( H `  v )
)  C_  U. ( H " om ) )
3129, 30syl 16 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  C_  U. ( H " om ) )
3231sseld 3349 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
( aleph `  z )  e.  ( aleph `  ( H `  v ) )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3320, 32syld 43 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3433rexlimiv 2826 . . . . . . . . . . . 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 2499 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  U. ( H " om )  <->  z  e.  (
aleph `  z ) ) )
37 eleq2 2499 . . . . . . . . . . . 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 172 . . . . . . . . 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 4593 . . . . . . . . . 10  |-  ( z  e.  On  ->  Ord  z )
438onordi 4688 . . . . . . . . . 10  |-  Ord  ( aleph `  z )
44 ordtri4 4620 . . . . . . . . . 10  |-  ( ( Ord  z  /\  Ord  ( aleph `  z )
)  ->  ( z  =  ( aleph `  z
)  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4542, 43, 44sylancl 645 . . . . . . . . 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 2447 . . . . . . . 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 2443 . . . . 5  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  ->  U. ( H " om )  =  z )
5251fveq2d 5734 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  ( aleph `  z
) )
53 eqeq2 2447 . . . . 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 2827 . 2  |-  ( E. z  e.  On  U. ( H " om )  =  ( aleph `  z
)  ->  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )
)
572, 6, 56mp2b 10 1  |-  ( aleph ` 
U. ( H " om ) )  =  U. ( H " om )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   U.cuni 4017   Ord word 4582   Oncon0 4583   suc csuc 4585   omcom 4847   ran crn 4881    |` cres 4882   "cima 4883   Fun wfun 5450    Fn wfn 5451   ` cfv 5456   reccrdg 6669   cardccrd 7824   alephcale 7825
This theorem is referenced by:  alephfp2  7992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-har 7528  df-card 7828  df-aleph 7829
  Copyright terms: Public domain W3C validator