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Theorem alephfp 7751
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 7752 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 7750 . 2  |-  U. ( H " om )  e. 
ran  aleph
3 isinfcard 7735 . . 3  |-  ( ( om  C_  U. ( H " om )  /\  ( card `  U. ( H
" om ) )  =  U. ( H
" om ) )  <->  U. ( H " om )  e.  ran  aleph )
4 cardalephex 7733 . . . 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 7731 . . . . . . . . 9  |-  ( z  e.  On  ->  z  C_  ( aleph `  z )
)
8 alephon 7712 . . . . . . . . . . 11  |-  ( aleph `  z )  e.  On
98onirri 4515 . . . . . . . . . 10  |-  -.  ( aleph `  z )  e.  ( aleph `  z )
10 frfnom 6463 . . . . . . . . . . . . . 14  |-  ( rec ( aleph ,  om )  |` 
om )  Fn  om
111fneq1i 5354 . . . . . . . . . . . . . 14  |-  ( H  Fn  om  <->  ( rec ( aleph ,  om )  |` 
om )  Fn  om )
1210, 11mpbir 200 . . . . . . . . . . . . 13  |-  H  Fn  om
13 fnfun 5357 . . . . . . . . . . . . 13  |-  ( H  Fn  om  ->  Fun  H )
14 eluniima 5792 . . . . . . . . . . . . 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 7743 . . . . . . . . . . . . . . . 16  |-  ran  aleph  C_  On
171alephfplem3 7749 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( H `  v )  e.  ran  aleph )
1816, 17sseldi 3191 . . . . . . . . . . . . . . 15  |-  ( v  e.  om  ->  ( H `  v )  e.  On )
19 alephord2i 7720 . . . . . . . . . . . . . . 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 7748 . . . . . . . . . . . . . . . . 17  |-  ( v  e.  om  ->  ( H `  suc  v )  =  ( aleph `  ( H `  v )
) )
22 peano2 4692 . . . . . . . . . . . . . . . . . 18  |-  ( v  e.  om  ->  suc  v  e.  om )
23 fnfvelrn 5678 . . . . . . . . . . . . . . . . . . . 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 5378 . . . . . . . . . . . . . . . . . . . 20  |-  ( H  Fn  om  ->  ( H " om )  =  ran  H )
2612, 25ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( H
" om )  =  ran  H
2724, 26syl6eleqr 2387 . . . . . . . . . . . . . . . . . 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 2371 . . . . . . . . . . . . . . . 16  |-  ( v  e.  om  ->  ( aleph `  ( H `  v ) )  e.  ( H " om ) )
30 elssuni 3871 . . . . . . . . . . . . . . . 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 3192 . . . . . . . . . . . . . 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 2674 . . . . . . . . . . . 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 2357 . . . . . . . . . . . 12  |-  ( U. ( H " om )  =  ( aleph `  z
)  ->  ( z  e.  U. ( H " om )  <->  z  e.  (
aleph `  z ) ) )
37 eleq2 2357 . . . . . . . . . . . 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 4418 . . . . . . . . . 10  |-  ( z  e.  On  ->  Ord  z )
438onordi 4513 . . . . . . . . . 10  |-  Ord  ( aleph `  z )
44 ordtri4 4445 . . . . . . . . . 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 2305 . . . . . . . 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 2301 . . . . 5  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  ->  U. ( H " om )  =  z )
5251fveq2d 5545 . . . 4  |-  ( ( z  e.  On  /\  U. ( H " om )  =  ( aleph `  z ) )  -> 
( aleph `  U. ( H
" om ) )  =  ( aleph `  z
) )
53 eqeq2 2305 . . . . 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 2675 . 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   U.cuni 3843   Ord word 4407   Oncon0 4408   suc csuc 4410   omcom 4672   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271   reccrdg 6438   cardccrd 7584   alephcale 7585
This theorem is referenced by:  alephfp2  7752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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