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Theorem cflecard 8089
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 8083 . . 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 3780 . . . . . 6  |-  { (
card `  A ) }  =  { x  |  x  =  ( card `  A ) }
3 ssid 3327 . . . . . . . . 9  |-  A  C_  A
4 ssid 3327 . . . . . . . . . . 11  |-  z  C_  z
5 sseq2 3330 . . . . . . . . . . . 12  |-  ( w  =  z  ->  (
z  C_  w  <->  z  C_  z ) )
65rspcev 3012 . . . . . . . . . . 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 2731 . . . . . . . . 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 5687 . . . . . . . . . . 11  |-  ( y  =  A  ->  ( card `  y )  =  ( card `  A
) )
1110eqeq2d 2415 . . . . . . . . . 10  |-  ( y  =  A  ->  (
x  =  ( card `  y )  <->  x  =  ( card `  A )
) )
12 sseq1 3329 . . . . . . . . . . 11  |-  ( y  =  A  ->  (
y  C_  A  <->  A  C_  A
) )
13 rexeq 2865 . . . . . . . . . . . 12  |-  ( y  =  A  ->  ( E. w  e.  y 
z  C_  w  <->  E. w  e.  A  z  C_  w ) )
1413ralbidv 2686 . . . . . . . . . . 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 2997 . . . . . . . 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 3376 . . . . . 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 3352 . . . . 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 4031 . . . . 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 5701 . . . . 5  |-  ( card `  A )  e.  _V
2423intsn 4046 . . . 4  |-  |^| { (
card `  A ) }  =  ( card `  A )
2522, 24syl6sseq 3354 . . 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 3342 . 2  |-  ( A  e.  On  ->  ( cf `  A )  C_  ( card `  A )
)
27 cff 8084 . . . . . 6  |-  cf : On
--> On
2827fdmi 5555 . . . . 5  |-  dom  cf  =  On
2928eleq2i 2468 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
30 ndmfv 5714 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3129, 30sylnbir 299 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
32 0ss 3616 . . 3  |-  (/)  C_  ( card `  A )
3331, 32syl6eqss 3358 . 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 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    C_ wss 3280   (/)c0 3588   {csn 3774   |^|cint 4010   Oncon0 4541   dom cdm 4837   ` cfv 5413   cardccrd 7778   cfccf 7780
This theorem is referenced by:  cfle  8090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-card 7782  df-cf 7784
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