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Theorem alephsing 8450
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 8283). 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 8240 . . . . . . 7  |-  aleph  Fn  On
2 fnfun 5513 . . . . . . 7  |-  ( aleph  Fn  On  ->  Fun  aleph )
31, 2ax-mp 5 . . . . . 6  |-  Fun  aleph
4 simpl 457 . . . . . 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 663 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  e. 
_V )
7 limelon 4787 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  On )
8 onss 6407 . . . . . . . 8  |-  ( A  e.  On  ->  A  C_  On )
97, 8syl 16 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  C_  On )
10 fnssres 5529 . . . . . . 7  |-  ( (
aleph  Fn  On  /\  A  C_  On )  ->  ( aleph 
|`  A )  Fn  A )
111, 9, 10sylancr 663 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A )  Fn  A )
12 fvres 5709 . . . . . . . . . . 11  |-  ( y  e.  A  ->  (
( aleph  |`  A ) `  y )  =  (
aleph `  y ) )
1312adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  =  ( aleph `  y )
)
14 alephord2i 8252 . . . . . . . . . . 11  |-  ( A  e.  On  ->  (
y  e.  A  -> 
( aleph `  y )  e.  ( aleph `  A )
) )
1514imp 429 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( aleph `  y )  e.  ( aleph `  A )
)
1613, 15eqeltrd 2517 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  e.  ( aleph `  A )
)
177, 16sylan 471 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\ 
Lim  A )  /\  y  e.  A )  ->  ( ( aleph  |`  A ) `
 y )  e.  ( aleph `  A )
)
1817ralrimiva 2804 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )
19 fnfvrnss 5876 . . . . . . 7  |-  ( ( ( aleph  |`  A )  Fn  A  /\  A. y  e.  A  ( ( aleph 
|`  A ) `  y )  e.  (
aleph `  A ) )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A ) )
2011, 18, 19syl2anc 661 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ran  ( aleph  |`  A )  C_  ( aleph `  A )
)
21 df-f 5427 . . . . . 6  |-  ( (
aleph  |`  A ) : A --> ( aleph `  A
)  <->  ( ( aleph  |`  A )  Fn  A  /\  ran  ( aleph  |`  A ) 
C_  ( aleph `  A
) ) )
2211, 20, 21sylanbrc 664 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph 
|`  A ) : A --> ( aleph `  A
) )
23 alephsmo 8277 . . . . . 6  |-  Smo  aleph
24 fndm 5515 . . . . . . . 8  |-  ( aleph  Fn  On  ->  dom  aleph  =  On )
251, 24ax-mp 5 . . . . . . 7  |-  dom  aleph  =  On
267, 25syl6eleqr 2534 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A  e.  dom  aleph )
27 smores 6818 . . . . . 6  |-  ( ( Smo  aleph  /\  A  e.  dom  aleph )  ->  Smo  ( aleph  |`  A ) )
2823, 26, 27sylancr 663 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  Smo  ( aleph  |`  A ) )
29 alephlim 8242 . . . . . . . 8  |-  ( ( A  e.  _V  /\  Lim  A )  ->  ( aleph `  A )  = 
U_ y  e.  A  ( aleph `  y )
)
3029eleq2d 2510 . . . . . . 7  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  <->  x  e.  U_ y  e.  A  (
aleph `  y ) ) )
31 eliun 4180 . . . . . . . 8  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  <->  E. y  e.  A  x  e.  ( aleph `  y ) )
32 alephon 8244 . . . . . . . . . 10  |-  ( aleph `  y )  e.  On
3332onelssi 4832 . . . . . . . . 9  |-  ( x  e.  ( aleph `  y
)  ->  x  C_  ( aleph `  y ) )
3433reximi 2828 . . . . . . . 8  |-  ( E. y  e.  A  x  e.  ( aleph `  y
)  ->  E. y  e.  A  x  C_  ( aleph `  y ) )
3531, 34sylbi 195 . . . . . . 7  |-  ( x  e.  U_ y  e.  A  ( aleph `  y
)  ->  E. y  e.  A  x  C_  ( aleph `  y ) )
3630, 35syl6bi 228 . . . . . 6  |-  ( ( A  e.  _V  /\  Lim  A )  ->  (
x  e.  ( aleph `  A )  ->  E. y  e.  A  x  C_  ( aleph `  y ) ) )
3736ralrimiv 2803 . . . . 5  |-  ( ( A  e.  _V  /\  Lim  A )  ->  A. x  e.  ( aleph `  A ) E. y  e.  A  x  C_  ( aleph `  y
) )
38 feq1 5547 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( f : A --> ( aleph `  A
)  <->  ( aleph  |`  A ) : A --> ( aleph `  A ) ) )
39 smoeq 6816 . . . . . . . 8  |-  ( f  =  ( aleph  |`  A )  ->  ( Smo  f  <->  Smo  ( aleph  |`  A ) ) )
40 fveq1 5695 . . . . . . . . . . . 12  |-  ( f  =  ( aleph  |`  A )  ->  ( f `  y )  =  ( ( aleph  |`  A ) `  y ) )
4140, 12sylan9eq 2495 . . . . . . . . . . 11  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
f `  y )  =  ( aleph `  y
) )
4241sseq2d 3389 . . . . . . . . . 10  |-  ( ( f  =  ( aleph  |`  A )  /\  y  e.  A )  ->  (
x  C_  ( f `  y )  <->  x  C_  ( aleph `  y ) ) )
4342rexbidva 2737 . . . . . . . . 9  |-  ( f  =  ( aleph  |`  A )  ->  ( E. y  e.  A  x  C_  (
f `  y )  <->  E. y  e.  A  x 
C_  ( aleph `  y
) ) )
4443ralbidv 2740 . . . . . . . 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 1289 . . . . . . 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 3063 . . . . . 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 429 . . . . 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 1220 . . . 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 8244 . . . . 5  |-  ( aleph `  A )  e.  On
50 cfcof 8448 . . . . 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 663 . . . 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 435 . 2  |-  ( Lim 
A  ->  ( A  e.  _V  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) ) )
54 cf0 8425 . . 3  |-  ( cf `  (/) )  =  (/)
55 fvprc 5690 . . . 4  |-  ( -.  A  e.  _V  ->  (
aleph `  A )  =  (/) )
5655fveq2d 5700 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  (/) ) )
57 fvprc 5690 . . 3  |-  ( -.  A  e.  _V  ->  ( cf `  A )  =  (/) )
5854, 56, 573eqtr4a 2501 . 2  |-  ( -.  A  e.  _V  ->  ( cf `  ( aleph `  A ) )  =  ( cf `  A
) )
5953, 58pm2.61d1 159 1  |-  ( Lim 
A  ->  ( cf `  ( aleph `  A )
)  =  ( cf `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   (/)c0 3642   U_ciun 4176   Oncon0 4724   Lim wlim 4725   dom cdm 4845   ran crn 4846    |` cres 4847   Fun wfun 5417    Fn wfn 5418   -->wf 5419   ` cfv 5423   Smo wsmo 6811   alephcale 8111   cfccf 8112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-smo 6812  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-oi 7729  df-har 7778  df-card 8114  df-aleph 8115  df-cf 8116  df-acn 8117
This theorem is referenced by:  alephom  8754  winafp  8869
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