MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardval3 Structured version   Unicode version

Theorem cardval3 7831
Description: An alternative definition of the value of  ( card `  A
) that does not require AC to prove. (Contributed by Mario Carneiro, 7-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
cardval3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Distinct variable group:    x, A

Proof of Theorem cardval3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2956 . 2  |-  ( A  e.  dom  card  ->  A  e.  _V )
2 isnum2 7824 . . . 4  |-  ( A  e.  dom  card  <->  E. x  e.  On  x  ~~  A
)
3 rabn0 3639 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  E. x  e.  On  x  ~~  A )
4 intex 4348 . . . 4  |-  ( { x  e.  On  |  x  ~~  A }  =/=  (/)  <->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
52, 3, 43bitr2i 265 . . 3  |-  ( A  e.  dom  card  <->  |^| { x  e.  On  |  x  ~~  A }  e.  _V )
65biimpi 187 . 2  |-  ( A  e.  dom  card  ->  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )
7 breq2 4208 . . . . 5  |-  ( y  =  A  ->  (
x  ~~  y  <->  x  ~~  A ) )
87rabbidv 2940 . . . 4  |-  ( y  =  A  ->  { x  e.  On  |  x  ~~  y }  =  {
x  e.  On  |  x  ~~  A } )
98inteqd 4047 . . 3  |-  ( y  =  A  ->  |^| { x  e.  On  |  x  ~~  y }  =  |^| { x  e.  On  |  x  ~~  A } )
10 df-card 7818 . . 3  |-  card  =  ( y  e.  _V  |->  |^|
{ x  e.  On  |  x  ~~  y } )
119, 10fvmptg 5796 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  On  |  x  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { x  e.  On  |  x  ~~  A }
)
121, 6, 11syl2anc 643 1  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { x  e.  On  |  x  ~~  A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701   _Vcvv 2948   (/)c0 3620   |^|cint 4042   class class class wbr 4204   Oncon0 4573   dom cdm 4870   ` cfv 5446    ~~ cen 7098   cardccrd 7814
This theorem is referenced by:  cardid2  7832  oncardval  7834  cardidm  7838  cardne  7844  cardval  8413
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-pr 4395  ax-un 4693
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-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 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-en 7102  df-card 7818
  Copyright terms: Public domain W3C validator