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Theorem alephsing 8145
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 7978). 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
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfnon 7935 . . . . . . 7  |-  aleph  Fn  On
2 fnfun 5533 . . . . . . 7  |-  ( aleph  Fn  On  ->  Fun  aleph )
31, 2ax-mp 8 . . . . . 6  |-  Fun  aleph
4 simpl 444 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  _V )
5 resfunexg 5948 . . . . . 6  |-  ( ( Fun  aleph  /\  A  e.  _V )  ->  ( aleph  |`  A )  e.  _V )
63, 4, 5sylancr 645 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  e. 
_V )
7 limelon 4636 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
8 onss 4762 . . . . . . . 8  |-  ( A  e.  On  ->  A  C_  On )
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  C_  On )
10 fnssres 5549 . . . . . . 7  |-  ( (
aleph  Fn  On  /\  A  C_  On )  ->  ( aleph 
|`  A )  Fn  A )
111, 9, 10sylancr 645 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  Fn  A )
12 fvres 5736 . . . . . . . . . . 11  |-  ( y  e.  A  ->  (
( aleph  |`  A ) `  y )  =  (
aleph `  y ) )
1312adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  =  ( aleph `  y )
)
14 alephord2i 7947 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
1514imp 419 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( aleph `  y )  e.  ( aleph `  A )
)
1613, 15eqeltrd 2509 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  e.  ( aleph `  A )
)
177, 16sylan 458 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\ 
Lim  A )  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  e.  ( aleph `  A )
)
1817ralrimiva 2781 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )
19 fnfvrnss 5887 . . . . . . 7  |-  ( ( ( aleph  |`  A )  Fn  A  /\  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A ) )
2011, 18, 19syl2anc 643 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A )
)
21 df-f 5449 . . . . . 6  |-  ( (
aleph  |`  A ) : A --> ( aleph `  A
)  <->  ( ( aleph  |`  A )  Fn  A  /\  ran  ( aleph  |`  A ) 
C_  ( aleph `  A
) ) )
2211, 20, 21sylanbrc 646 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A ) : A --> ( aleph `  A
) )
23 alephsmo 7972 . . . . . 6  |-  Smo  aleph
24 fndm 5535 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
251, 24ax-mp 8 . . . . . . 7  |-  dom  aleph  =  On
267, 25syl6eleqr 2526 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  dom  aleph )
27 smores 6605 . . . . . 6  |-  ( ( Smo  aleph  /\  A  e.  dom  aleph )  ->  Smo  ( aleph  |`  A ) )
2823, 26, 27sylancr 645 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  Smo  ( aleph  |`  A ) )
29 alephlim 7937 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ y  e.  A  ( aleph `  y )
)
3029eleq2d 2502 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  <->  x  e.  U_ y  e.  A  (
aleph `  y ) ) )
31 eliun 4089 . . . . . . . 8  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  <->  E. y  e.  A  x  e.  ( aleph `  y ) )
32 alephon 7939 . . . . . . . . . 10  |-  ( aleph `  y )  e.  On
3332onelssi 4681 . . . . . . . . 9  |-  ( x  e.  ( aleph `  y
)  ->  x  C_  ( aleph `  y ) )
3433reximi 2805 . . . . . . . 8  |-  ( E. y  e.  A  x  e.  ( aleph `  y
)  ->  E. y  e.  A  x  C_  ( aleph `  y ) )
3531, 34sylbi 188 . . . . . . 7  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  ->  E. y  e.  A  x  C_  ( aleph `  y ) )
3630, 35syl6bi 220 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  ->  E. y  e.  A  x  C_  ( aleph `  y ) ) )
3736ralrimiv 2780 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y
) )
38 feq1 5567 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( f : A --> ( aleph `  A
)  <->  ( aleph  |`  A ) : A --> ( aleph `  A ) ) )
39 smoeq 6603 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( Smo  f  <->  Smo  ( aleph  |`  A ) ) )
40 fveq1 5718 . . . . . . . . . . . 12  |-  ( f  =  ( aleph  |`  A )  ->  ( f `  y )  =  ( ( aleph  |`  A ) `  y ) )
4140, 12sylan9eq 2487 . . . . . . . . . . 11  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
f `  y )  =  ( aleph `  y
) )
4241sseq2d 3368 . . . . . . . . . 10  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
x  C_  ( f `  y )  <->  x  C_  ( aleph `  y ) ) )
4342rexbidva 2714 . . . . . . . . 9  |-  ( f  =  ( aleph  |`  A )  ->  ( E. y  e.  A  x  C_  (
f `  y )  <->  E. y  e.  A  x 
C_  ( aleph `  y
) ) )
4443ralbidv 2717 . . . . . . . 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 1254 . . . . . . 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 3029 . . . . . 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 419 . . . . 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 1186 . . . 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 7939 . . . . 5  |-  ( aleph `  A )  e.  On
50 cfcof 8143 . . . . 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 645 . . . 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 15 . . 3  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( cf `  ( aleph `  A
) )  =  ( cf `  A ) )
5352expcom 425 . 2  |-  ( Lim 
A  ->  ( A  e.  _V  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) ) )
54 cf0 8120 . . 3  |-  ( cf `  (/) )  =  (/)
55 fvprc 5713 . . . 4  |-  ( -.  A  e.  _V  ->  (
aleph `  A )  =  (/) )
5655fveq2d 5723 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  (/) ) )
57 fvprc 5713 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  A )  =  (/) )
5854, 56, 573eqtr4a 2493 . 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 3    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   (/)c0 3620   U_ciun 4085   Oncon0 4573   Lim wlim 4574   dom cdm 4869   ran crn 4870    |` cres 4871   Fun wfun 5439    Fn wfn 5440   -->wf 5441   ` cfv 5445   Smo wsmo 6598   alephcale 7812   cfccf 7813
This theorem is referenced by:  alephom  8449  winafp  8561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-smo 6599  df-recs 6624  df-rdg 6659  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816  df-cf 7817  df-acn 7818
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