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

Theorem cardval 8912
Description: The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8363 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 8841 . 2  |-  ( A  e.  _V  ->  A  e.  dom  card )
3 cardval3 8324 . 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 1398    e. wcel 1823   {crab 2808   _Vcvv 3106   |^|cint 4271   class class class wbr 4439   Oncon0 4867   dom cdm 4988   ` cfv 5570    ~~ cen 7506   cardccrd 8307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-ac2 8834
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-recs 7034  df-en 7510  df-card 8311  df-ac 8488
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator