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Theorem alephsing 8187
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 8020). 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 7977 . . . . . . 7  |-  aleph  Fn  On
2 fnfun 5571 . . . . . . 7  |-  ( aleph  Fn  On  ->  Fun  aleph )
31, 2ax-mp 5 . . . . . 6  |-  Fun  aleph
4 simpl 445 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  _V )
5 resfunexg 5986 . . . . . 6  |-  ( ( Fun  aleph  /\  A  e.  _V )  ->  ( aleph  |`  A )  e.  _V )
63, 4, 5sylancr 646 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  e. 
_V )
7 limelon 4673 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
8 onss 4800 . . . . . . . 8  |-  ( A  e.  On  ->  A  C_  On )
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  C_  On )
10 fnssres 5587 . . . . . . 7  |-  ( (
aleph  Fn  On  /\  A  C_  On )  ->  ( aleph 
|`  A )  Fn  A )
111, 9, 10sylancr 646 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  Fn  A )
12 fvres 5774 . . . . . . . . . . 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 7989 . . . . . . . . . . 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 2516 . . . . . . . . 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 2795 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )
19 fnfvrnss 5925 . . . . . . 7  |-  ( ( ( aleph  |`  A )  Fn  A  /\  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A ) )
2011, 18, 19syl2anc 644 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A )
)
21 df-f 5487 . . . . . 6  |-  ( (
aleph  |`  A ) : A --> ( aleph `  A
)  <->  ( ( aleph  |`  A )  Fn  A  /\  ran  ( aleph  |`  A ) 
C_  ( aleph `  A
) ) )
2211, 20, 21sylanbrc 647 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A ) : A --> ( aleph `  A
) )
23 alephsmo 8014 . . . . . 6  |-  Smo  aleph
24 fndm 5573 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
251, 24ax-mp 5 . . . . . . 7  |-  dom  aleph  =  On
267, 25syl6eleqr 2533 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  dom  aleph )
27 smores 6643 . . . . . 6  |-  ( ( Smo  aleph  /\  A  e.  dom  aleph )  ->  Smo  ( aleph  |`  A ) )
2823, 26, 27sylancr 646 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  Smo  ( aleph  |`  A ) )
29 alephlim 7979 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ y  e.  A  ( aleph `  y )
)
3029eleq2d 2509 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  <->  x  e.  U_ y  e.  A  (
aleph `  y ) ) )
31 eliun 4121 . . . . . . . 8  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  <->  E. y  e.  A  x  e.  ( aleph `  y ) )
32 alephon 7981 . . . . . . . . . 10  |-  ( aleph `  y )  e.  On
3332onelssi 4719 . . . . . . . . 9  |-  ( x  e.  ( aleph `  y
)  ->  x  C_  ( aleph `  y ) )
3433reximi 2819 . . . . . . . 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 2794 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y
) )
38 feq1 5605 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( f : A --> ( aleph `  A
)  <->  ( aleph  |`  A ) : A --> ( aleph `  A ) ) )
39 smoeq 6641 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( Smo  f  <->  Smo  ( aleph  |`  A ) ) )
40 fveq1 5756 . . . . . . . . . . . 12  |-  ( f  =  ( aleph  |`  A )  ->  ( f `  y )  =  ( ( aleph  |`  A ) `  y ) )
4140, 12sylan9eq 2494 . . . . . . . . . . 11  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
f `  y )  =  ( aleph `  y
) )
4241sseq2d 3362 . . . . . . . . . 10  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
x  C_  ( f `  y )  <->  x  C_  ( aleph `  y ) ) )
4342rexbidva 2728 . . . . . . . . 9  |-  ( f  =  ( aleph  |`  A )  ->  ( E. y  e.  A  x  C_  (
f `  y )  <->  E. y  e.  A  x 
C_  ( aleph `  y
) ) )
4443ralbidv 2731 . . . . . . . 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 1255 . . . . . . 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 3043 . . . . . 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 1187 . . . 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 7981 . . . . 5  |-  ( aleph `  A )  e.  On
50 cfcof 8185 . . . . 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 646 . . . 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 426 . 2  |-  ( Lim 
A  ->  ( A  e.  _V  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) ) )
54 cf0 8162 . . 3  |-  ( cf `  (/) )  =  (/)
55 fvprc 5751 . . . 4  |-  ( -.  A  e.  _V  ->  (
aleph `  A )  =  (/) )
5655fveq2d 5761 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  (/) ) )
57 fvprc 5751 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  A )  =  (/) )
5854, 56, 573eqtr4a 2500 . 2  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  A
) )
5953, 58pm2.61d1 154 1  |-  ( Lim 
A  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712   _Vcvv 2962    C_ wss 3306   (/)c0 3613   U_ciun 4117   Oncon0 4610   Lim wlim 4611   dom cdm 4907   ran crn 4908    |` cres 4909   Fun wfun 5477    Fn wfn 5478   -->wf 5479   ` cfv 5483   Smo wsmo 6636   alephcale 7854   cfccf 7855
This theorem is referenced by:  alephom  8491  winafp  8603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-smo 6637  df-recs 6662  df-rdg 6697  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-oi 7508  df-har 7555  df-card 7857  df-aleph 7858  df-cf 7859  df-acn 7860
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