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Theorem cardid2 8237
Description: Any numerable set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardid2  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )

Proof of Theorem cardid2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardval3 8236 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
2 ssrab2 3548 . . . 4  |-  { y  e.  On  |  y 
~~  A }  C_  On
3 fvex 5812 . . . . . 6  |-  ( card `  A )  e.  _V
41, 3syl6eqelr 2551 . . . . 5  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )
5 intex 4559 . . . . 5  |-  ( { y  e.  On  | 
y  ~~  A }  =/=  (/)  <->  |^| { y  e.  On  |  y  ~~  A }  e.  _V )
64, 5sylibr 212 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =/=  (/) )
7 onint 6519 . . . 4  |-  ( ( { y  e.  On  |  y  ~~  A }  C_  On  /\  { y  e.  On  |  y 
~~  A }  =/=  (/) )  ->  |^| { y  e.  On  |  y 
~~  A }  e.  { y  e.  On  | 
y  ~~  A }
)
82, 6, 7sylancr 663 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  e.  { y  e.  On  |  y  ~~  A }
)
91, 8eqeltrd 2542 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  e.  { y  e.  On  |  y  ~~  A }
)
10 breq1 4406 . . . 4  |-  ( y  =  ( card `  A
)  ->  ( y  ~~  A  <->  ( card `  A
)  ~~  A )
)
1110elrab 3224 . . 3  |-  ( (
card `  A )  e.  { y  e.  On  |  y  ~~  A }  <->  ( ( card `  A
)  e.  On  /\  ( card `  A )  ~~  A ) )
1211simprbi 464 . 2  |-  ( (
card `  A )  e.  { y  e.  On  |  y  ~~  A }  ->  ( card `  A
)  ~~  A )
139, 12syl 16 1  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    =/= wne 2648   {crab 2803   _Vcvv 3078    C_ wss 3439   (/)c0 3748   |^|cint 4239   class class class wbr 4403   Oncon0 4830   dom cdm 4951   ` cfv 5529    ~~ cen 7420   cardccrd 8219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-en 7424  df-card 8223
This theorem is referenced by:  isnum3  8238  oncardid  8240  cardidm  8243  ficardom  8245  ficardid  8246  cardnn  8247  cardnueq0  8248  carden2a  8250  carden2b  8251  carddomi2  8254  sdomsdomcardi  8255  cardsdomelir  8257  cardsdomel  8258  infxpidm2  8297  dfac8b  8315  numdom  8322  alephnbtwn2  8356  alephsucdom  8363  infenaleph  8375  dfac12r  8429  cardacda  8481  pwsdompw  8487  cff1  8541  cfflb  8542  cflim2  8546  cfss  8548  cfslb  8549  domtriomlem  8725  cardid  8825  cardidg  8826  carden  8829  sdomsdomcard  8838  hargch  8954  gch2  8956  hashkf  12225
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