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Theorem cardval 8822
Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8273 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypothesis
Ref Expression
cardval.1  |-  A  e. 
_V
Assertion
Ref Expression
cardval  |-  ( card `  A )  =  |^| { x  e.  On  |  x  ~~  A }
Distinct variable group:    x, A

Proof of Theorem cardval
StepHypRef Expression
1 cardval.1 . 2  |-  A  e. 
_V
2 numth3 8751 . 2  |-  ( A  e.  _V  ->  A  e.  dom  card )
3 cardval3 8234 . 2  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
41, 2, 3mp2b 10 1  |-  ( card `  A )  =  |^| { x  e.  On  |  x  ~~  A }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078   |^|cint 4237   class class class wbr 4401   Oncon0 4828   dom cdm 4949   ` cfv 5527    ~~ cen 7418   cardccrd 8217
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-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-ac2 8744
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-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-recs 6943  df-en 7422  df-card 8221  df-ac 8398
This theorem is referenced by: (None)
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