MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardidm Unicode version

Theorem cardidm 7476
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
StepHypRef Expression
1 cardid2 7470 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 ensym 6796 . . . . . . . 8  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
31, 2syl 17 . . . . . . 7  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
4 entr 6798 . . . . . . . 8  |-  ( ( y  ~~  A  /\  A  ~~  ( card `  A
) )  ->  y  ~~  ( card `  A
) )
54expcom 426 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  ( y  ~~  A  ->  y  ~~  ( card `  A )
) )
63, 5syl 17 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  A  -> 
y  ~~  ( card `  A ) ) )
7 entr 6798 . . . . . . . 8  |-  ( ( y  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  y  ~~  A )
87expcom 426 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  ( y 
~~  ( card `  A
)  ->  y  ~~  A ) )
91, 8syl 17 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  ( card `  A )  ->  y  ~~  A ) )
106, 9impbid 185 . . . . 5  |-  ( A  e.  dom  card  ->  ( y  ~~  A  <->  y  ~~  ( card `  A )
) )
1110rabbidv 2719 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =  { y  e.  On  |  y  ~~  ( card `  A ) } )
1211inteqd 3765 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  =  |^| { y  e.  On  |  y  ~~  ( card `  A ) } )
13 cardval3 7469 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
14 cardon 7461 . . . 4  |-  ( card `  A )  e.  On
15 oncardval 7472 . . . 4  |-  ( (
card `  A )  e.  On  ->  ( card `  ( card `  A
) )  =  |^| { y  e.  On  | 
y  ~~  ( card `  A ) } )
1614, 15mp1i 13 . . 3  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  = 
|^| { y  e.  On  |  y  ~~  ( card `  A ) } )
1712, 13, 163eqtr4rd 2296 . 2  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  =  ( card `  A
) )
18 card0 7475 . . 3  |-  ( card `  (/) )  =  (/)
19 ndmfv 5405 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2019fveq2d 5381 . . 3  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  (/) ) )
2118, 20, 193eqtr4a 2311 . 2  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  A
) )
2217, 21pm2.61i 158 1  |-  ( card `  ( card `  A
) )  =  (
card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621   {crab 2512   (/)c0 3362   |^|cint 3760   class class class wbr 3920   Oncon0 4285   dom cdm 4580   ` cfv 4592    ~~ cen 6746   cardccrd 7452
This theorem is referenced by:  oncard  7477  cardlim  7489  cardiun  7499  alephnbtwn2  7583  infenaleph  7602  dfac12k  7657  pwsdompw  7714  cardcf  7762  cfeq0  7766  cfflb  7769  alephval2  8074  cfpwsdom  8086  gch2  8181  tskcard  8283  hashcard  11227
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-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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-rab 2516  df-v 2729  df-sbc 2922  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-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-we 4247  df-ord 4288  df-on 4289  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-er 6546  df-en 6750  df-card 7456
  Copyright terms: Public domain W3C validator