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Theorem brtxpsd 25744
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 4216 . . 3  |-  ( A ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) ) B  <->  <. A ,  B >.  e.  ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) )
2 opex 4430 . . . . 5  |-  <. A ,  B >.  e.  _V
32elrn 5113 . . . 4  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) )  <->  E. x  x (
( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. )
4 brsymdif 25678 . . . . . 6  |-  ( x ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x ( _V  (x)  _E  ) <. A ,  B >.  <->  x
( R  (x)  _V ) <. A ,  B >. ) )
5 brv 25727 . . . . . . . . 9  |-  x _V A
6 vex 2961 . . . . . . . . . 10  |-  x  e. 
_V
7 brtxpsd.1 . . . . . . . . . 10  |-  A  e. 
_V
8 brtxpsd.2 . . . . . . . . . 10  |-  B  e. 
_V
96, 7, 8brtxp 25730 . . . . . . . . 9  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
( x _V A  /\  x  _E  B
) )
105, 9mpbiran 886 . . . . . . . 8  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  _E  B )
118epelc 4499 . . . . . . . 8  |-  ( x  _E  B  <->  x  e.  B )
1210, 11bitri 242 . . . . . . 7  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  e.  B )
13 brv 25727 . . . . . . . 8  |-  x _V B
146, 7, 8brtxp 25730 . . . . . . . 8  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
( x R A  /\  x _V B
) )
1513, 14mpbiran2 887 . . . . . . 7  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
x R A )
1612, 15bibi12i 308 . . . . . 6  |-  ( ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x ( R  (x)  _V ) <. A ,  B >. )  <->  ( x  e.  B  <->  x R A ) )
174, 16xchbinx 303 . . . . 5  |-  ( x ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x  e.  B  <->  x R A ) )
1817exbii 1593 . . . 4  |-  ( E. x  x ( ( _V  (x)  _E  )(++) ( R  (x)  _V )
) <. A ,  B >.  <->  E. x  -.  (
x  e.  B  <->  x R A ) )
193, 18bitri 242 . . 3  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )(++) ( R  (x)  _V ) )  <->  E. x  -.  (
x  e.  B  <->  x R A ) )
20 exnal 1584 . . 3  |-  ( E. x  -.  ( x  e.  B  <->  x R A )  <->  -.  A. x
( x  e.  B  <->  x R A ) )
211, 19, 203bitrri 265 . 2  |-  ( -. 
A. x ( x  e.  B  <->  x R A )  <->  A ran  ( ( _V  (x)  _E  )(++) ( R  (x)  _V ) ) B )
2221con1bii 323 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 178   A.wal 1550   E.wex 1551    e. wcel 1726   _Vcvv 2958   <.cop 3819   class class class wbr 4215    _E cep 4495   ran crn 4882  (++)csymdif 25667    (x) ctxp 25679
This theorem is referenced by:  brtxpsd2  25745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-eprel 4497  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-1st 6352  df-2nd 6353  df-symdif 25668  df-txp 25703
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