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Theorem cardval3ex 7162
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 6724 . . . 4  |-  ( x 
~~  A  ->  (
x  e.  _V  /\  A  e.  _V )
)
21simprd 113 . . 3  |-  ( x 
~~  A  ->  A  e.  _V )
32rexlimivw 2583 . 2  |-  ( E. x  e.  On  x  ~~  A  ->  A  e. 
_V )
4 breq1 3992 . . . 4  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
54cbvrexv 2697 . . 3  |-  ( E. y  e.  On  y  ~~  A  <->  E. x  e.  On  x  ~~  A )
6 intexrabim 4139 . . 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 3993 . . . . 5  |-  ( z  =  A  ->  (
y  ~~  z  <->  y  ~~  A ) )
98rabbidv 2719 . . . 4  |-  ( z  =  A  ->  { y  e.  On  |  y 
~~  z }  =  { y  e.  On  |  y  ~~  A }
)
109inteqd 3836 . . 3  |-  ( z  =  A  ->  |^| { y  e.  On  |  y 
~~  z }  =  |^| { y  e.  On  |  y  ~~  A }
)
11 df-card 7157 . . 3  |-  card  =  ( z  e.  _V  |->  |^|
{ y  e.  On  |  y  ~~  z } )
1210, 11fvmptg 5572 . 2  |-  ( ( A  e.  _V  /\  |^|
{ y  e.  On  |  y  ~~  A }  e.  _V )  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
133, 7, 12syl2anc 409 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 1348    e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   |^|cint 3831   class class class wbr 3989   Oncon0 4348   ` cfv 5198    ~~ cen 6716   cardccrd 7156
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-en 6719  df-card 7157
This theorem is referenced by:  oncardval  7163  carden2bex  7166
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