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Theorem alephsing 7786
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 7619). 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 7576 . . . . . . 7  |-  aleph  Fn  On
2 fnfun 5198 . . . . . . 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 5589 . . . . . 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 4348 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
8 onss 4473 . . . . . . . 8  |-  ( A  e.  On  ->  A  C_  On )
97, 8syl 17 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  C_  On )
10 fnssres 5214 . . . . . . 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 5394 . . . . . . . . . . 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 7588 . . . . . . . . . . 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 2327 . . . . . . . . 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 2588 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )
19 fnfvrnss 5539 . . . . . . 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 4604 . . . . . 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 7613 . . . . . 6  |-  Smo  aleph
24 fndm 5200 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
251, 24ax-mp 10 . . . . . . 7  |-  dom  aleph  =  On
267, 25syl6eleqr 2344 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  dom  aleph )
27 smores 6255 . . . . . 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 7578 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ y  e.  A  ( aleph `  y )
)
3029eleq2d 2320 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  <->  x  e.  U_ y  e.  A  (
aleph `  y ) ) )
31 eliun 3807 . . . . . . . 8  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  <->  E. y  e.  A  x  e.  ( aleph `  y ) )
32 alephon 7580 . . . . . . . . . 10  |-  ( aleph `  y )  e.  On
3332onelssi 4392 . . . . . . . . 9  |-  ( x  e.  ( aleph `  y
)  ->  x  C_  ( aleph `  y ) )
3433reximi 2612 . . . . . . . 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 2587 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y
) )
38 feq1 5232 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( f : A --> ( aleph `  A
)  <->  ( aleph  |`  A ) : A --> ( aleph `  A ) ) )
39 smoeq 6253 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( Smo  f  <->  Smo  ( aleph  |`  A ) ) )
40 fveq1 5376 . . . . . . . . . . . 12  |-  ( f  =  ( aleph  |`  A )  ->  ( f `  y )  =  ( ( aleph  |`  A ) `  y ) )
4140, 12sylan9eq 2305 . . . . . . . . . . 11  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
f `  y )  =  ( aleph `  y
) )
4241sseq2d 3127 . . . . . . . . . 10  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
x  C_  ( f `  y )  <->  x  C_  ( aleph `  y ) ) )
4342rexbidva 2524 . . . . . . . . 9  |-  ( f  =  ( aleph  |`  A )  ->  ( E. y  e.  A  x  C_  (
f `  y )  <->  E. y  e.  A  x 
C_  ( aleph `  y
) ) )
4443ralbidv 2527 . . . . . . . 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 ) ) ) )
4645cla4egv 2806 . . . . . 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 7580 . . . . 5  |-  ( aleph `  A )  e.  On
50 cfcof 7784 . . . . 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 7761 . . 3  |-  ( cf `  (/) )  =  (/)
55 fvprc 5374 . . . 4  |-  ( -.  A  e.  _V  ->  (
aleph `  A )  =  (/) )
5655fveq2d 5381 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  (/) ) )
57 fvprc 5374 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  A )  =  (/) )
5854, 56, 573eqtr4a 2311 . 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 1537    = wceq 1619    e. wcel 1621   A.wral 2509   E.wrex 2510   _Vcvv 2727    C_ wss 3078   (/)c0 3362   U_ciun 3803   Oncon0 4285   Lim wlim 4286   dom cdm 4580   ran crn 4581    |` cres 4582   Fun wfun 4586    Fn wfn 4587   -->wf 4588   ` cfv 4592   Smo wsmo 6248   alephcale 7453   cfccf 7454
This theorem is referenced by:  alephom  8087  winafp  8199
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
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 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-smo 6249  df-recs 6274  df-rdg 6309  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-har 7156  df-card 7456  df-aleph 7457  df-cf 7458  df-acn 7459
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