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

Theorem brcart 25778
Description: Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcart.1  |-  A  e. 
_V
brcart.2  |-  B  e. 
_V
brcart.3  |-  C  e. 
_V
Assertion
Ref Expression
brcart  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )

Proof of Theorem brcart
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4428 . 2  |-  <. A ,  B >.  e.  _V
2 brcart.3 . 2  |-  C  e. 
_V
3 df-cart 25710 . 2  |- Cart  =  ( ( ( _V  X.  _V )  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (pprod (  _E  ,  _E  )  (x)  _V ) ) )
4 brcart.1 . . . 4  |-  A  e. 
_V
5 brcart.2 . . . 4  |-  B  e. 
_V
64, 5opelvv 4925 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4910 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 888 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 3anass 941 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  _E  A  /\  z  _E  B ) ) )
104epelc 4497 . . . . . . 7  |-  ( y  _E  A  <->  y  e.  A )
115epelc 4497 . . . . . . 7  |-  ( z  _E  B  <->  z  e.  B )
1210, 11anbi12i 680 . . . . . 6  |-  ( ( y  _E  A  /\  z  _E  B )  <->  ( y  e.  A  /\  z  e.  B )
)
1312anbi2i 677 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  (
y  _E  A  /\  z  _E  B )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
149, 13bitri 242 . . . 4  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B ) ) )
15142exbii 1594 . . 3  |-  ( E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
16 vex 2960 . . . 4  |-  x  e. 
_V
1716, 4, 5brpprod3b 25733 . . 3  |-  ( xpprod (  _E  ,  _E  ) <. A ,  B >.  <->  E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B ) )
18 elxp 4896 . . 3  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
1915, 17, 183bitr4ri 271 . 2  |-  ( x  e.  ( A  X.  B )  <->  xpprod (  _E  ,  _E  ) <. A ,  B >. )
201, 2, 3, 8, 19brtxpsd3 25742 1  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2957   <.cop 3818   class class class wbr 4213    _E cep 4493    X. cxp 4877  pprodcpprod 25676  Cartccart 25686
This theorem is referenced by:  brimg  25783  brrestrict  25795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-eprel 4495  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463  df-1st 6350  df-2nd 6351  df-symdif 25664  df-txp 25699  df-pprod 25700  df-cart 25710
  Copyright terms: Public domain W3C validator