Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brtxpsd Structured version   Unicode version

Theorem brtxpsd 30647
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 4418 . . 3  |-  ( A ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) B  <->  <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) )
2 opex 4678 . . . . 5  |-  <. A ,  B >.  e.  _V
32elrn 5087 . . . 4  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) )  <->  E. x  x ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) ) <. A ,  B >. )
4 brsymdif 4474 . . . . . 6  |-  ( x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x ( _V  (x)  _E  ) <. A ,  B >.  <->  x
( R  (x)  _V ) <. A ,  B >. ) )
5 brv 30630 . . . . . . . . 9  |-  x _V A
6 vex 3081 . . . . . . . . . 10  |-  x  e. 
_V
7 brtxpsd.1 . . . . . . . . . 10  |-  A  e. 
_V
8 brtxpsd.2 . . . . . . . . . 10  |-  B  e. 
_V
96, 7, 8brtxp 30633 . . . . . . . . 9  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
( x _V A  /\  x  _E  B
) )
105, 9mpbiran 926 . . . . . . . 8  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  _E  B )
118epelc 4759 . . . . . . . 8  |-  ( x  _E  B  <->  x  e.  B )
1210, 11bitri 252 . . . . . . 7  |-  ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x  e.  B )
13 brv 30630 . . . . . . . 8  |-  x _V B
146, 7, 8brtxp 30633 . . . . . . . 8  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
( x R A  /\  x _V B
) )
1513, 14mpbiran2 927 . . . . . . 7  |-  ( x ( R  (x)  _V ) <. A ,  B >.  <-> 
x R A )
1612, 15bibi12i 316 . . . . . 6  |-  ( ( x ( _V  (x)  _E  ) <. A ,  B >.  <-> 
x ( R  (x)  _V ) <. A ,  B >. )  <->  ( x  e.  B  <->  x R A ) )
174, 16xchbinx 311 . . . . 5  |-  ( x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  -.  ( x  e.  B  <->  x R A ) )
1817exbii 1712 . . . 4  |-  ( E. x  x ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) <. A ,  B >. 
<->  E. x  -.  (
x  e.  B  <->  x R A ) )
193, 18bitri 252 . . 3  |-  ( <. A ,  B >.  e. 
ran  ( ( _V 
(x)  _E  )  /_\  ( R 
(x)  _V ) )  <->  E. x  -.  ( x  e.  B  <->  x R A ) )
20 exnal 1695 . . 3  |-  ( E. x  -.  ( x  e.  B  <->  x R A )  <->  -.  A. x
( x  e.  B  <->  x R A ) )
211, 19, 203bitrri 275 . 2  |-  ( -. 
A. x ( x  e.  B  <->  x R A )  <->  A ran  ( ( _V  (x)  _E  )  /_\  ( R  (x)  _V ) ) B )
2221con1bii 332 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 187   A.wal 1435   E.wex 1659    e. wcel 1867   _Vcvv 3078    /_\ csymdif 3689   <.cop 3999   class class class wbr 4417    _E cep 4755   ran crn 4847    (x) ctxp 30582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-symdif 3690  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-eprel 4757  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-fo 5599  df-fv 5601  df-1st 6799  df-2nd 6800  df-txp 30606
This theorem is referenced by:  brtxpsd2  30648
  Copyright terms: Public domain W3C validator