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Theorem brtxpsd 30732
Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brtxpsd.1  |-  A  e. 
_V
brtxpsd.2  |-  B  e. 
_V
Assertion
Ref Expression
brtxpsd  |-  ( -.  A ran  ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) B  <->  A. x
( x  e.  B  <->  x R A ) )
Distinct variable groups:    x, A    x, B    x, R

Proof of Theorem brtxpsd
StepHypRef Expression
1 df-br 4396 . . 3  |-  ( A ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) B  <->  <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) )
2 opex 4664 . . . . 5  |-  <. A ,  B >.  e.  _V
32elrn 5081 . . . 4  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) )  <->  E. x  x ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) <. A ,  B >. )
4 brsymdif 4452 . . . . . 6  |-  ( x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x ( _V  (x)  _E  ) <. A ,  B >.  <->  x
( R  (x)  _V ) <. A ,  B >. ) )
5 brv 30715 . . . . . . . . 9  |-  x _V A
6 vex 3034 . . . . . . . . . 10  |-  x  e. 
_V
7 brtxpsd.1 . . . . . . . . . 10  |-  A  e. 
_V
8 brtxpsd.2 . . . . . . . . . 10  |-  B  e. 
_V
96, 7, 8brtxp 30718 . . . . . . . . 9  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
( x _V A  /\  x  _E  B
) )
105, 9mpbiran 932 . . . . . . . 8  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  _E  B )
118epelc 4752 . . . . . . . 8  |-  ( x  _E  B  <->  x  e.  B )
1210, 11bitri 257 . . . . . . 7  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  e.  B )
13 brv 30715 . . . . . . . 8  |-  x _V B
146, 7, 8brtxp 30718 . . . . . . . 8  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
( x R A  /\  x _V B
) )
1513, 14mpbiran2 933 . . . . . . 7  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
x R A )
1612, 15bibi12i 322 . . . . . 6  |-  ( ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x ( R  (x)  _V ) <. A ,  B >. )  <->  ( x  e.  B  <->  x R A ) )
174, 16xchbinx 317 . . . . 5  |-  ( x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x  e.  B  <->  x R A ) )
1817exbii 1726 . . . 4  |-  ( E. x  x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  E. x  -.  (
x  e.  B  <->  x R A ) )
193, 18bitri 257 . . 3  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) )  <->  E. x  -.  ( x  e.  B  <->  x R A ) )
20 exnal 1707 . . 3  |-  ( E. x  -.  ( x  e.  B  <->  x R A )  <->  -.  A. x
( x  e.  B  <->  x R A ) )
211, 19, 203bitrri 280 . 2  |-  ( -. 
A. x ( x  e.  B  <->  x R A )  <->  A ran  ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) B )
2221con1bii 338 1  |-  ( -.  A ran  ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) B  <->  A. x
( x  e.  B  <->  x R A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189   A.wal 1450   E.wex 1671    e. wcel 1904   _Vcvv 3031    /_\ csymdif 3653   <.cop 3965   class class class wbr 4395    _E cep 4748   ran crn 4840    (x) ctxp 30667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691
This theorem is referenced by:  brtxpsd2  30733
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