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Theorem cflecard 8123
Description: Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cflecard  |-  ( cf `  A )  C_  ( card `  A )

Proof of Theorem cflecard
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 8117 . . 3  |-  ( A  e.  On  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
2 df-sn 3812 . . . . . 6  |-  { (
card `  A ) }  =  { x  |  x  =  ( card `  A ) }
3 ssid 3359 . . . . . . . . 9  |-  A  C_  A
4 ssid 3359 . . . . . . . . . . 11  |-  z  C_  z
5 sseq2 3362 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
z  C_  w  <->  z  C_  z ) )
65rspcev 3044 . . . . . . . . . . 11  |-  ( ( z  e.  A  /\  z  C_  z )  ->  E. w  e.  A  z  C_  w )
74, 6mpan2 653 . . . . . . . . . 10  |-  ( z  e.  A  ->  E. w  e.  A  z  C_  w )
87rgen 2763 . . . . . . . . 9  |-  A. z  e.  A  E. w  e.  A  z  C_  w
93, 8pm3.2i 442 . . . . . . . 8  |-  ( A 
C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w )
10 fveq2 5720 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( card `  y )  =  ( card `  A
) )
1110eqeq2d 2446 . . . . . . . . . 10  |-  ( y  =  A  ->  (
x  =  ( card `  y )  <->  x  =  ( card `  A )
) )
12 sseq1 3361 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  C_  A  <->  A  C_  A
) )
13 rexeq 2897 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( E. w  e.  y 
z  C_  w  <->  E. w  e.  A  z  C_  w ) )
1413ralbidv 2717 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( A. z  e.  A  E. w  e.  y 
z  C_  w  <->  A. z  e.  A  E. w  e.  A  z  C_  w ) )
1512, 14anbi12d 692 . . . . . . . . . 10  |-  ( y  =  A  ->  (
( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )  <->  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) ) )
1611, 15anbi12d 692 . . . . . . . . 9  |-  ( y  =  A  ->  (
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
)  <->  ( x  =  ( card `  A
)  /\  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) ) ) )
1716spcegv 3029 . . . . . . . 8  |-  ( A  e.  On  ->  (
( x  =  (
card `  A )  /\  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) )  ->  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) ) )
189, 17mpan2i 659 . . . . . . 7  |-  ( A  e.  On  ->  (
x  =  ( card `  A )  ->  E. y
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
) ) )
1918ss2abdv 3408 . . . . . 6  |-  ( A  e.  On  ->  { x  |  x  =  ( card `  A ) } 
C_  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
202, 19syl5eqss 3384 . . . . 5  |-  ( A  e.  On  ->  { (
card `  A ) }  C_  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
21 intss 4063 . . . . 5  |-  ( { ( card `  A
) }  C_  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  |^| { ( card `  A ) } )
2220, 21syl 16 . . . 4  |-  ( A  e.  On  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  |^| { ( card `  A ) } )
23 fvex 5734 . . . . 5  |-  ( card `  A )  e.  _V
2423intsn 4078 . . . 4  |-  |^| { (
card `  A ) }  =  ( card `  A )
2522, 24syl6sseq 3386 . . 3  |-  ( A  e.  On  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  ( card `  A
) )
261, 25eqsstrd 3374 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  ( card `  A )
)
27 cff 8118 . . . . . 6  |-  cf : On
--> On
2827fdmi 5588 . . . . 5  |-  dom  cf  =  On
2928eleq2i 2499 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
30 ndmfv 5747 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3129, 30sylnbir 299 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
32 0ss 3648 . . 3  |-  (/)  C_  ( card `  A )
3331, 32syl6eqss 3390 . 2  |-  ( -.  A  e.  On  ->  ( cf `  A ) 
C_  ( card `  A
) )
3426, 33pm2.61i 158 1  |-  ( cf `  A )  C_  ( card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   {csn 3806   |^|cint 4042   Oncon0 4573   dom cdm 4870   ` cfv 5446   cardccrd 7812   cfccf 7814
This theorem is referenced by:  cfle  8124
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-rab 2706  df-v 2950  df-sbc 3154  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-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-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-card 7816  df-cf 7818
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