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Theorem alephsing 7897
Description: The cofinality of a limit aleph is the same as the cofinality of its argument, so if  ( aleph `  A )  <  A, then  ( aleph `  A
) is singular. Conversely, if  ( aleph `  A ) is regular (i.e. weakly inaccessible), then  ( aleph `  A )  =  A, so  A has to be rather large (see alephfp 7730). Proposition 11.13 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
Assertion
Ref Expression
alephsing  |-  ( Lim 
A  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) )

Proof of Theorem alephsing
StepHypRef Expression
1 alephfnon 7687 . . . . . . 7  |-  aleph  Fn  On
2 fnfun 5306 . . . . . . 7  |-  ( aleph  Fn  On  ->  Fun  aleph )
31, 2ax-mp 10 . . . . . 6  |-  Fun  aleph
4 simpl 445 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  _V )
5 resfunexg 5698 . . . . . 6  |-  ( ( Fun  aleph  /\  A  e.  _V )  ->  ( aleph  |`  A )  e.  _V )
63, 4, 5sylancr 647 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  e. 
_V )
7 limelon 4454 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
8 onss 4581 . . . . . . . 8  |-  ( A  e.  On  ->  A  C_  On )
97, 8syl 17 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  C_  On )
10 fnssres 5322 . . . . . . 7  |-  ( (
aleph  Fn  On  /\  A  C_  On )  ->  ( aleph 
|`  A )  Fn  A )
111, 9, 10sylancr 647 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  Fn  A )
12 fvres 5502 . . . . . . . . . . 11  |-  ( y  e.  A  ->  (
( aleph  |`  A ) `  y )  =  (
aleph `  y ) )
1312adantl 454 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  =  ( aleph `  y )
)
14 alephord2i 7699 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
1514imp 420 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( aleph `  y )  e.  ( aleph `  A )
)
1613, 15eqeltrd 2358 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  e.  ( aleph `  A )
)
177, 16sylan 459 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\ 
Lim  A )  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  e.  ( aleph `  A )
)
1817ralrimiva 2627 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )
19 fnfvrnss 5648 . . . . . . 7  |-  ( ( ( aleph  |`  A )  Fn  A  /\  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A ) )
2011, 18, 19syl2anc 645 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A )
)
21 df-f 5225 . . . . . 6  |-  ( (
aleph  |`  A ) : A --> ( aleph `  A
)  <->  ( ( aleph  |`  A )  Fn  A  /\  ran  ( aleph  |`  A ) 
C_  ( aleph `  A
) ) )
2211, 20, 21sylanbrc 648 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A ) : A --> ( aleph `  A
) )
23 alephsmo 7724 . . . . . 6  |-  Smo  aleph
24 fndm 5308 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
251, 24ax-mp 10 . . . . . . 7  |-  dom  aleph  =  On
267, 25syl6eleqr 2375 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  dom  aleph )
27 smores 6364 . . . . . 6  |-  ( ( Smo  aleph  /\  A  e.  dom  aleph )  ->  Smo  ( aleph  |`  A ) )
2823, 26, 27sylancr 647 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  Smo  ( aleph  |`  A ) )
29 alephlim 7689 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ y  e.  A  ( aleph `  y )
)
3029eleq2d 2351 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  <->  x  e.  U_ y  e.  A  (
aleph `  y ) ) )
31 eliun 3910 . . . . . . . 8  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  <->  E. y  e.  A  x  e.  ( aleph `  y ) )
32 alephon 7691 . . . . . . . . . 10  |-  ( aleph `  y )  e.  On
3332onelssi 4500 . . . . . . . . 9  |-  ( x  e.  ( aleph `  y
)  ->  x  C_  ( aleph `  y ) )
3433reximi 2651 . . . . . . . 8  |-  ( E. y  e.  A  x  e.  ( aleph `  y
)  ->  E. y  e.  A  x  C_  ( aleph `  y ) )
3531, 34sylbi 189 . . . . . . 7  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  ->  E. y  e.  A  x  C_  ( aleph `  y ) )
3630, 35syl6bi 221 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  ->  E. y  e.  A  x  C_  ( aleph `  y ) ) )
3736ralrimiv 2626 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y
) )
38 feq1 5340 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( f : A --> ( aleph `  A
)  <->  ( aleph  |`  A ) : A --> ( aleph `  A ) ) )
39 smoeq 6362 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( Smo  f  <->  Smo  ( aleph  |`  A ) ) )
40 fveq1 5484 . . . . . . . . . . . 12  |-  ( f  =  ( aleph  |`  A )  ->  ( f `  y )  =  ( ( aleph  |`  A ) `  y ) )
4140, 12sylan9eq 2336 . . . . . . . . . . 11  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
f `  y )  =  ( aleph `  y
) )
4241sseq2d 3207 . . . . . . . . . 10  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
x  C_  ( f `  y )  <->  x  C_  ( aleph `  y ) ) )
4342rexbidva 2561 . . . . . . . . 9  |-  ( f  =  ( aleph  |`  A )  ->  ( E. y  e.  A  x  C_  (
f `  y )  <->  E. y  e.  A  x 
C_  ( aleph `  y
) ) )
4443ralbidv 2564 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( f `  y )  <->  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y
) ) )
4538, 39, 443anbi123d 1257 . . . . . . 7  |-  ( f  =  ( aleph  |`  A )  ->  ( ( f : A --> ( aleph `  A )  /\  Smo  f  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( f `  y ) )  <->  ( ( aleph 
|`  A ) : A --> ( aleph `  A
)  /\  Smo  ( aleph  |`  A )  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y ) ) ) )
4645spcegv 2870 . . . . . 6  |-  ( (
aleph  |`  A )  e. 
_V  ->  ( ( (
aleph  |`  A ) : A --> ( aleph `  A
)  /\  Smo  ( aleph  |`  A )  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y ) )  ->  E. f ( f : A --> ( aleph `  A )  /\  Smo  f  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( f `  y ) ) ) )
4746imp 420 . . . . 5  |-  ( ( ( aleph  |`  A )  e. 
_V  /\  ( ( aleph 
|`  A ) : A --> ( aleph `  A
)  /\  Smo  ( aleph  |`  A )  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y ) ) )  ->  E. f
( f : A --> ( aleph `  A )  /\  Smo  f  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  (
f `  y )
) )
486, 22, 28, 37, 47syl13anc 1189 . . . 4  |-  ( ( A  e.  _V  /\  Lim  A )  ->  E. f
( f : A --> ( aleph `  A )  /\  Smo  f  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  (
f `  y )
) )
49 alephon 7691 . . . . 5  |-  ( aleph `  A )  e.  On
50 cfcof 7895 . . . . 5  |-  ( ( ( aleph `  A )  e.  On  /\  A  e.  On )  ->  ( E. f ( f : A --> ( aleph `  A
)  /\  Smo  f  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  (
f `  y )
)  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) ) )
5149, 7, 50sylancr 647 . . . 4  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( E. f ( f : A --> ( aleph `  A
)  /\  Smo  f  /\  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  (
f `  y )
)  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) ) )
5248, 51mpd 16 . . 3  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( cf `  ( aleph `  A
) )  =  ( cf `  A ) )
5352expcom 426 . 2  |-  ( Lim 
A  ->  ( A  e.  _V  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) ) )
54 cf0 7872 . . 3  |-  ( cf `  (/) )  =  (/)
55 fvprc 5482 . . . 4  |-  ( -.  A  e.  _V  ->  (
aleph `  A )  =  (/) )
5655fveq2d 5489 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  (/) ) )
57 fvprc 5482 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  A )  =  (/) )
5854, 56, 573eqtr4a 2342 . 2  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  A
) )
5953, 58pm2.61d1 153 1  |-  ( Lim 
A  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939   E.wex 1533    = wceq 1628    e. wcel 1688   A.wral 2544   E.wrex 2545   _Vcvv 2789    C_ wss 3153   (/)c0 3456   U_ciun 3906   Oncon0 4391   Lim wlim 4392   dom cdm 4688   ran crn 4689    |` cres 4690   Fun wfun 5215    Fn wfn 5216   -->wf 5217   ` cfv 5221   Smo wsmo 6357   alephcale 7564   cfccf 7565
This theorem is referenced by:  alephom  8202  winafp  8314
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
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 1534  df-nf 1537  df-sb 1636  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-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-smo 6358  df-recs 6383  df-rdg 6418  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-har 7267  df-card 7567  df-aleph 7568  df-cf 7569  df-acn 7570
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