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Theorem brtxpsd 30654
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 4402 . . 3  |-  ( A ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) B  <->  <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) )
2 opex 4663 . . . . 5  |-  <. A ,  B >.  e.  _V
32elrn 5074 . . . 4  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) )  <->  E. x  x ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) <. A ,  B >. )
4 brsymdif 4458 . . . . . 6  |-  ( x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x ( _V  (x)  _E  ) <. A ,  B >.  <->  x
( R  (x)  _V ) <. A ,  B >. ) )
5 brv 30637 . . . . . . . . 9  |-  x _V A
6 vex 3047 . . . . . . . . . 10  |-  x  e. 
_V
7 brtxpsd.1 . . . . . . . . . 10  |-  A  e. 
_V
8 brtxpsd.2 . . . . . . . . . 10  |-  B  e. 
_V
96, 7, 8brtxp 30640 . . . . . . . . 9  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
( x _V A  /\  x  _E  B
) )
105, 9mpbiran 928 . . . . . . . 8  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  _E  B )
118epelc 4746 . . . . . . . 8  |-  ( x  _E  B  <->  x  e.  B )
1210, 11bitri 253 . . . . . . 7  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  e.  B )
13 brv 30637 . . . . . . . 8  |-  x _V B
146, 7, 8brtxp 30640 . . . . . . . 8  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
( x R A  /\  x _V B
) )
1513, 14mpbiran2 929 . . . . . . 7  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
x R A )
1612, 15bibi12i 317 . . . . . 6  |-  ( ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x ( R  (x)  _V ) <. A ,  B >. )  <->  ( x  e.  B  <->  x R A ) )
174, 16xchbinx 312 . . . . 5  |-  ( x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x  e.  B  <->  x R A ) )
1817exbii 1717 . . . 4  |-  ( E. x  x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  E. x  -.  (
x  e.  B  <->  x R A ) )
193, 18bitri 253 . . 3  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) )  <->  E. x  -.  ( x  e.  B  <->  x R A ) )
20 exnal 1698 . . 3  |-  ( E. x  -.  ( x  e.  B  <->  x R A )  <->  -.  A. x
( x  e.  B  <->  x R A ) )
211, 19, 203bitrri 276 . 2  |-  ( -. 
A. x ( x  e.  B  <->  x R A )  <->  A ran  ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) B )
2221con1bii 333 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 188   A.wal 1441   E.wex 1662    e. wcel 1886   _Vcvv 3044    /_\ csymdif 3661   <.cop 3973   class class class wbr 4401    _E cep 4742   ran crn 4834    (x) ctxp 30589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-symdif 3662  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-eprel 4744  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6790  df-2nd 6791  df-txp 30613
This theorem is referenced by:  brtxpsd2  30655
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