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Theorem alephfp 8557
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 8558 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 8556 . 2  |-  U. ( H " om )  e. 
ran  aleph
3 isinfcard 8541 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  <->  U. ( H " om )  e.  ran  aleph )
4 cardalephex 8539 . . . 4  |-  ( om  C_  U. ( H " om )  ->  ( (
card `  U. ( H
" om ) )  =  U. ( H
" om )  <->  E. z  e.  On  U. ( H
" om )  =  ( aleph `  z )
) )
54biimpa 492 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  ->  E. z  e.  On  U. ( H " om )  =  ( aleph `  z ) )
63, 5sylbir 218 . 2  |-  ( U. ( H " om )  e.  ran  aleph  ->  E. z  e.  On  U. ( H
" om )  =  ( aleph `  z )
)
7 alephle 8537 . . . . . . . . 9  |-  ( z  e.  On  ->  z  C_  ( aleph `  z )
)
8 alephon 8518 . . . . . . . . . . 11  |-  ( aleph `  z )  e.  On
98onirri 5536 . . . . . . . . . 10  |-  -.  ( aleph `  z )  e.  ( aleph `  z )
10 frfnom 7170 . . . . . . . . . . . . . 14  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
111fneq1i 5680 . . . . . . . . . . . . . 14  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
1210, 11mpbir 214 . . . . . . . . . . . . 13  |-  H  Fn  om
13 fnfun 5683 . . . . . . . . . . . . 13  |-  ( H  Fn  om  ->  Fun  H )
14 eluniima 6173 . . . . . . . . . . . . 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 8549 . . . . . . . . . . . . . . . 16  |-  ran  aleph  C_  On
171alephfplem3 8555 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
1816, 17sseldi 3416 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( H `  v )  e.  On )
19 alephord2i 8526 . . . . . . . . . . . . . . 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 8554 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  =  ( aleph `  ( H `  v )
) )
22 peano2 6732 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  om  ->  suc  v  e.  om )
23 fnfvelrn 6034 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( H  Fn  om  /\  suc  v  e.  om )  ->  ( H `  suc  v )  e.  ran  H )
2412, 23mpan 684 . . . . . . . . . . . . . . . . . . 19  |-  ( suc  v  e.  om  ->  ( H `  suc  v
)  e.  ran  H
)
25 fnima 5704 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  Fn  om  ->  ( H " om )  =  ran  H )
2612, 25ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  ( H
" om )  =  ran  H
2724, 26syl6eleqr 2560 . . . . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  e.  ( H " om ) )
30 elssuni 4219 . . . . . . . . . . . . . . . 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 3417 . . . . . . . . . . . . . 14  |-  ( v  e.  om  ->  (
( aleph `  z )  e.  ( aleph `  ( H `  v ) )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3320, 32syld 44 . . . . . . . . . . . . 13  |-  ( v  e.  om  ->  (
z  e.  ( H `
 v )  -> 
( aleph `  z )  e.  U. ( H " om ) ) )
3433rexlimiv 2867 . . . . . . . . . . . 12  |-  ( E. v  e.  om  z  e.  ( H `  v
)  ->  ( aleph `  z )  e.  U. ( H " om )
)
3515, 34sylbi 200 . . . . . . . . . . 11  |-  ( z  e.  U. ( H
" om )  -> 
( aleph `  z )  e.  U. ( H " om ) )
36 eleq2 2538 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  U. ( H " om )  <->  z  e.  (
aleph `  z ) ) )
37 eleq2 2538 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( ( aleph `  z )  e. 
U. ( H " om )  <->  ( aleph `  z
)  e.  ( aleph `  z ) ) )
3836, 37imbi12d 327 . . . . . . . . . . 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 216 . . . . . . . . . 10  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  ( aleph `  z )  ->  ( aleph `  z )  e.  ( aleph `  z )
) )
409, 39mtoi 183 . . . . . . . . 9  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  -.  z  e.  ( aleph `  z )
)
417, 40anim12i 576 . . . . . . . 8  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  C_  ( aleph `  z )  /\  -.  z  e.  ( aleph `  z ) ) )
42 eloni 5440 . . . . . . . . . 10  |-  ( z  e.  On  ->  Ord  z )
438onordi 5534 . . . . . . . . . 10  |-  Ord  ( aleph `  z )
44 ordtri4 5467 . . . . . . . . . 10  |-  ( ( Ord  z  /\  Ord  ( aleph `  z )
)  ->  ( z  =  ( aleph `  z
)  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4542, 43, 44sylancl 675 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  =  ( aleph `  z )  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4645adantr 472 . . . . . . . 8  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  =  (
aleph `  z )  <->  ( z  C_  ( aleph `  z )  /\  -.  z  e.  (
aleph `  z ) ) ) )
4741, 46mpbird 240 . . . . . . 7  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
z  =  ( aleph `  z ) )
48 eqeq2 2482 . . . . . . . 8  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  =  U. ( H " om )  <->  z  =  (
aleph `  z ) ) )
4948adantl 473 . . . . . . 7  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( z  =  U. ( H " om )  <->  z  =  ( aleph `  z
) ) )
5047, 49mpbird 240 . . . . . 6  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
z  =  U. ( H " om ) )
5150eqcomd 2477 . . . . 5  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  ->  U. ( H " om )  =  z )
5251fveq2d 5883 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  ( aleph `  z
) )
53 eqeq2 2482 . . . . 5  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( ( aleph `  U. ( H
" om ) )  =  U. ( H
" om )  <->  ( aleph ` 
U. ( H " om ) )  =  (
aleph `  z ) ) )
5453adantl 473 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( ( aleph `  U. ( H " om )
)  =  U. ( H " om )  <->  ( aleph ` 
U. ( H " om ) )  =  (
aleph `  z ) ) )
5552, 54mpbird 240 . . 3  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  U. ( H
" om ) )
5655rexlimiva 2868 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757    C_ wss 3390   U.cuni 4190   ran crn 4840    |` cres 4841   "cima 4842   Ord word 5429   Oncon0 5430   suc csuc 5432   Fun wfun 5583    Fn wfn 5584   ` cfv 5589   omcom 6711   reccrdg 7145   cardccrd 8387   alephcale 8388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-har 8091  df-card 8391  df-aleph 8392
This theorem is referenced by:  alephfp2  8558
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