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Theorem dfint3 30705
Description: Quantifier-free definition of class intersection. (Contributed by Scott Fenton, 13-Apr-2018.)
Assertion
Ref Expression
dfint3  |-  |^| A  =  ( _V  \ 
( `' ( _V 
\  _E  ) " A ) )

Proof of Theorem dfint3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfint2 4251 . 2  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
2 ralnex 2869 . . . 4  |-  ( A. y  e.  A  -.  y `' ( _V  \  _E  ) x  <->  -.  E. y  e.  A  y `' ( _V  \  _E  )
x )
3 vex 3081 . . . . . . . . 9  |-  y  e. 
_V
4 vex 3081 . . . . . . . . 9  |-  x  e. 
_V
53, 4brcnv 5029 . . . . . . . 8  |-  ( y `' ( _V  \  _E  ) x  <->  x ( _V  \  _E  ) y )
6 brv 30630 . . . . . . . . 9  |-  x _V y
7 brdif 4468 . . . . . . . . 9  |-  ( x ( _V  \  _E  ) y  <->  ( x _V y  /\  -.  x  _E  y ) )
86, 7mpbiran 926 . . . . . . . 8  |-  ( x ( _V  \  _E  ) y  <->  -.  x  _E  y )
95, 8bitr2i 253 . . . . . . 7  |-  ( -.  x  _E  y  <->  y `' ( _V  \  _E  )
x )
109con1bii 332 . . . . . 6  |-  ( -.  y `' ( _V 
\  _E  ) x  <-> 
x  _E  y )
11 epel 4760 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
1210, 11bitr2i 253 . . . . 5  |-  ( x  e.  y  <->  -.  y `' ( _V  \  _E  ) x )
1312ralbii 2854 . . . 4  |-  ( A. y  e.  A  x  e.  y  <->  A. y  e.  A  -.  y `' ( _V 
\  _E  ) x )
14 eldif 3443 . . . . . 6  |-  ( x  e.  ( _V  \ 
( `' ( _V 
\  _E  ) " A ) )  <->  ( x  e.  _V  /\  -.  x  e.  ( `' ( _V 
\  _E  ) " A ) ) )
154, 14mpbiran 926 . . . . 5  |-  ( x  e.  ( _V  \ 
( `' ( _V 
\  _E  ) " A ) )  <->  -.  x  e.  ( `' ( _V 
\  _E  ) " A ) )
164elima 5185 . . . . 5  |-  ( x  e.  ( `' ( _V  \  _E  ) " A )  <->  E. y  e.  A  y `' ( _V  \  _E  )
x )
1715, 16xchbinx 311 . . . 4  |-  ( x  e.  ( _V  \ 
( `' ( _V 
\  _E  ) " A ) )  <->  -.  E. y  e.  A  y `' ( _V  \  _E  )
x )
182, 13, 173bitr4ri 281 . . 3  |-  ( x  e.  ( _V  \ 
( `' ( _V 
\  _E  ) " A ) )  <->  A. y  e.  A  x  e.  y )
1918abbi2i 2553 . 2  |-  ( _V 
\  ( `' ( _V  \  _E  ) " A ) )  =  { x  |  A. y  e.  A  x  e.  y }
201, 19eqtr4i 2452 1  |-  |^| A  =  ( _V  \ 
( `' ( _V 
\  _E  ) " A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1437    e. wcel 1867   {cab 2405   A.wral 2773   E.wrex 2774   _Vcvv 3078    \ cdif 3430   |^|cint 4249   class class class wbr 4417    _E cep 4755   `'ccnv 4845   "cima 4849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-int 4250  df-br 4418  df-opab 4477  df-eprel 4757  df-xp 4852  df-cnv 4854  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859
This theorem is referenced by: (None)
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