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Theorem cardval3ex 6803
Description: The value of  ( card `  A ). (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
cardval3ex  |-  ( E. x  e.  On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
Distinct variable group:    x, A, y

Proof of Theorem cardval3ex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 encv 6453 . . . 4  |-  ( x 
~~  A  ->  (
x  e.  _V  /\  A  e.  _V )
)
21simprd 112 . . 3  |-  ( x 
~~  A  ->  A  e.  _V )
32rexlimivw 2485 . 2  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
4 breq1 3846 . . . 4  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
54cbvrexv 2591 . . 3  |-  ( E. y  e.  On  y  ~~  A  <->  E. x  e.  On  x  ~~  A )
6 intexrabim 3987 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  _V )
75, 6sylbir 133 . 2  |-  ( E. x  e.  On  x  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  _V )
8 breq2 3847 . . . . 5  |-  ( z  =  A  ->  (
y  ~~  z  <->  y  ~~  A ) )
98rabbidv 2608 . . . 4  |-  ( z  =  A  ->  { y  e.  On  |  y 
~~  z }  =  { y  e.  On  |  y  ~~  A }
)
109inteqd 3691 . . 3  |-  ( z  =  A  ->  |^| { y  e.  On  |  y 
~~  z }  =  |^| { y  e.  On  |  y  ~~  A }
)
11 df-card 6798 . . 3  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
1210, 11fvmptg 5374 . 2  |-  ( ( A  e.  _V  /\  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
133, 7, 12syl2anc 403 1  |-  ( E. x  e.  On  x  ~~  A  ->  ( card `  A )  =  |^| { y  e.  On  | 
y  ~~  A }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   E.wrex 2360   {crab 2363   _Vcvv 2619   |^|cint 3686   class class class wbr 3843   Oncon0 4188   ` cfv 5010    ~~ cen 6445   cardccrd 6797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-iota 4975  df-fun 5012  df-fv 5018  df-en 6448  df-card 6798
This theorem is referenced by:  oncardval  6804  carden2bex  6807
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