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Theorem brtxpsd 24992
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 4103 . . 3  |-  ( A ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) ) B  <->  <. A ,  B >.  e.  ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) )
2 opex 4316 . . . . 5  |-  <. A ,  B >.  e.  _V
32elrn 4998 . . . 4  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) )  <->  E. x  x (
( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. )
4 brsymdif 24930 . . . . . 6  |-  ( x ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x ( _V  (x)  _E  ) <. A ,  B >.  <->  x
( R  (x)  _V ) <. A ,  B >. ) )
5 brv 24975 . . . . . . . . 9  |-  x _V A
6 vex 2867 . . . . . . . . . 10  |-  x  e. 
_V
7 brtxpsd.1 . . . . . . . . . 10  |-  A  e. 
_V
8 brtxpsd.2 . . . . . . . . . 10  |-  B  e. 
_V
96, 7, 8brtxp 24978 . . . . . . . . 9  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
( x _V A  /\  x  _E  B
) )
105, 9mpbiran 884 . . . . . . . 8  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  _E  B )
118epelc 4386 . . . . . . . 8  |-  ( x  _E  B  <->  x  e.  B )
1210, 11bitri 240 . . . . . . 7  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  e.  B )
13 brv 24975 . . . . . . . 8  |-  x _V B
146, 7, 8brtxp 24978 . . . . . . . 8  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
( x R A  /\  x _V B
) )
1513, 14mpbiran2 885 . . . . . . 7  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
x R A )
1612, 15bibi12i 306 . . . . . 6  |-  ( ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x ( R  (x)  _V ) <. A ,  B >. )  <->  ( x  e.  B  <->  x R A ) )
174, 16xchbinx 301 . . . . 5  |-  ( x ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x  e.  B  <->  x R A ) )
1817exbii 1582 . . . 4  |-  ( E. x  x ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) <. A ,  B >.  <->  E. x  -.  (
x  e.  B  <->  x R A ) )
193, 18bitri 240 . . 3  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) )  <->  E. x  -.  (
x  e.  B  <->  x R A ) )
20 exnal 1574 . . 3  |-  ( E. x  -.  ( x  e.  B  <->  x R A )  <->  -.  A. x
( x  e.  B  <->  x R A ) )
211, 19, 203bitrri 263 . 2  |-  ( -. 
A. x ( x  e.  B  <->  x R A )  <->  A ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) B )
2221con1bii 321 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 176   A.wal 1540   E.wex 1541    e. wcel 1710   _Vcvv 2864   <.cop 3719   class class class wbr 4102    _E cep 4382   ran crn 4769  (++)csymdif 24919    (x) ctxp 24931
This theorem is referenced by:  brtxpsd2  24993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-eprel 4384  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fo 5340  df-fv 5342  df-1st 6206  df-2nd 6207  df-symdif 24920  df-txp 24953
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