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Theorem cardidm 7835
Description: The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardidm  |-  ( card `  ( card `  A
) )  =  (
card `  A )

Proof of Theorem cardidm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardid2 7829 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7149 . . . . . . 7  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 entr 7150 . . . . . . . 8  |-  ( ( y  ~~  A  /\  A  ~~  ( card `  A
) )  ->  y  ~~  ( card `  A
) )
43expcom 425 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  ( y  ~~  A  ->  y  ~~  ( card `  A )
) )
52, 4syl 16 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  A  -> 
y  ~~  ( card `  A ) ) )
6 entr 7150 . . . . . . . 8  |-  ( ( y  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  y  ~~  A )
76expcom 425 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  ( y 
~~  ( card `  A
)  ->  y  ~~  A ) )
81, 7syl 16 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  ( card `  A )  ->  y  ~~  A ) )
95, 8impbid 184 . . . . 5  |-  ( A  e.  dom  card  ->  ( y  ~~  A  <->  y  ~~  ( card `  A )
) )
109rabbidv 2940 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =  { y  e.  On  |  y  ~~  ( card `  A ) } )
1110inteqd 4047 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  =  |^| { y  e.  On  |  y  ~~  ( card `  A ) } )
12 cardval3 7828 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
13 cardon 7820 . . . 4  |-  ( card `  A )  e.  On
14 oncardval 7831 . . . 4  |-  ( (
card `  A )  e.  On  ->  ( card `  ( card `  A
) )  =  |^| { y  e.  On  | 
y  ~~  ( card `  A ) } )
1513, 14mp1i 12 . . 3  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  = 
|^| { y  e.  On  |  y  ~~  ( card `  A ) } )
1611, 12, 153eqtr4rd 2478 . 2  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  =  ( card `  A
) )
17 card0 7834 . . 3  |-  ( card `  (/) )  =  (/)
18 ndmfv 5746 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
1918fveq2d 5723 . . 3  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  (/) ) )
2017, 19, 183eqtr4a 2493 . 2  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  A
) )
2116, 20pm2.61i 158 1  |-  ( card `  ( card `  A
) )  =  (
card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701   (/)c0 3620   |^|cint 4042   class class class wbr 4204   Oncon0 4573   dom cdm 4869   ` cfv 5445    ~~ cen 7097   cardccrd 7811
This theorem is referenced by:  oncard  7836  cardlim  7848  cardiun  7858  alephnbtwn2  7942  infenaleph  7961  dfac12k  8016  pwsdompw  8073  cardcf  8121  cfeq0  8125  cfflb  8128  alephval2  8436  cfpwsdom  8448  gch2  8543  tskcard  8645  hashcard  11626
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-er 6896  df-en 7101  df-card 7815
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