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Theorem brtxpsd 25652
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 4177 . . 3  |-  ( A ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) ) B  <->  <. A ,  B >.  e.  ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) )
2 opex 4391 . . . . 5  |-  <. A ,  B >.  e.  _V
32elrn 5073 . . . 4  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) )  <->  E. x  x (
( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. )
4 brsymdif 25590 . . . . . 6  |-  ( x ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x ( _V  (x)  _E  ) <. A ,  B >.  <->  x
( R  (x)  _V ) <. A ,  B >. ) )
5 brv 25635 . . . . . . . . 9  |-  x _V A
6 vex 2923 . . . . . . . . . 10  |-  x  e. 
_V
7 brtxpsd.1 . . . . . . . . . 10  |-  A  e. 
_V
8 brtxpsd.2 . . . . . . . . . 10  |-  B  e. 
_V
96, 7, 8brtxp 25638 . . . . . . . . 9  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
( x _V A  /\  x  _E  B
) )
105, 9mpbiran 885 . . . . . . . 8  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  _E  B )
118epelc 4460 . . . . . . . 8  |-  ( x  _E  B  <->  x  e.  B )
1210, 11bitri 241 . . . . . . 7  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  e.  B )
13 brv 25635 . . . . . . . 8  |-  x _V B
146, 7, 8brtxp 25638 . . . . . . . 8  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
( x R A  /\  x _V B
) )
1513, 14mpbiran2 886 . . . . . . 7  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
x R A )
1612, 15bibi12i 307 . . . . . 6  |-  ( ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x ( R  (x)  _V ) <. A ,  B >. )  <->  ( x  e.  B  <->  x R A ) )
174, 16xchbinx 302 . . . . 5  |-  ( x ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x  e.  B  <->  x R A ) )
1817exbii 1589 . . . 4  |-  ( E. x  x ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) <. A ,  B >.  <->  E. x  -.  (
x  e.  B  <->  x R A ) )
193, 18bitri 241 . . 3  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) )  <->  E. x  -.  (
x  e.  B  <->  x R A ) )
20 exnal 1580 . . 3  |-  ( E. x  -.  ( x  e.  B  <->  x R A )  <->  -.  A. x
( x  e.  B  <->  x R A ) )
211, 19, 203bitrri 264 . 2  |-  ( -. 
A. x ( x  e.  B  <->  x R A )  <->  A ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) B )
2221con1bii 322 1  |-  ( -.  A ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) B  <->  A. x
( x  e.  B  <->  x R A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1546   E.wex 1547    e. wcel 1721   _Vcvv 2920   <.cop 3781   class class class wbr 4176    _E cep 4456   ran crn 4842  (++)csymdif 25579    (x) ctxp 25591
This theorem is referenced by:  brtxpsd2  25653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-eprel 4458  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fo 5423  df-fv 5425  df-1st 6312  df-2nd 6313  df-symdif 25580  df-txp 25613
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