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Theorem cardval3ex 7007
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 6606 . . . 4  |-  ( x 
~~  A  ->  (
x  e.  _V  /\  A  e.  _V )
)
21simprd 113 . . 3  |-  ( x 
~~  A  ->  A  e.  _V )
32rexlimivw 2520 . 2  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
4 breq1 3900 . . . 4  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
54cbvrexv 2630 . . 3  |-  ( E. y  e.  On  y  ~~  A  <->  E. x  e.  On  x  ~~  A )
6 intexrabim 4046 . . 3  |-  ( E. y  e.  On  y  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  _V )
75, 6sylbir 134 . 2  |-  ( E. x  e.  On  x  ~~  A  ->  |^| { y  e.  On  |  y 
~~  A }  e.  _V )
8 breq2 3901 . . . . 5  |-  ( z  =  A  ->  (
y  ~~  z  <->  y  ~~  A ) )
98rabbidv 2647 . . . 4  |-  ( z  =  A  ->  { y  e.  On  |  y 
~~  z }  =  { y  e.  On  |  y  ~~  A }
)
109inteqd 3744 . . 3  |-  ( z  =  A  ->  |^| { y  e.  On  |  y 
~~  z }  =  |^| { y  e.  On  |  y  ~~  A }
)
11 df-card 7002 . . 3  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
1210, 11fvmptg 5463 . 2  |-  ( ( A  e.  _V  /\  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
133, 7, 12syl2anc 406 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 1314    e. wcel 1463   E.wrex 2392   {crab 2395   _Vcvv 2658   |^|cint 3739   class class class wbr 3897   Oncon0 4253   ` cfv 5091    ~~ cen 6598   cardccrd 7001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-en 6601  df-card 7002
This theorem is referenced by:  oncardval  7008  carden2bex  7011
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