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Theorem brcart 24856
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 4274 . 2  |-  <. A ,  B >.  e.  _V
2 brcart.3 . 2  |-  C  e. 
_V
3 df-cart 24791 . 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 4772 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4757 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 886 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 3anass 938 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  _E  A  /\  z  _E  B ) ) )
104epelc 4344 . . . . . . 7  |-  ( y  _E  A  <->  y  e.  A )
115epelc 4344 . . . . . . 7  |-  ( z  _E  B  <->  z  e.  B )
1210, 11anbi12i 678 . . . . . 6  |-  ( ( y  _E  A  /\  z  _E  B )  <->  ( y  e.  A  /\  z  e.  B )
)
1312anbi2i 675 . . . . 5  |-  ( ( x  =  <. y ,  z >.  /\  (
y  _E  A  /\  z  _E  B )
)  <->  ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
149, 13bitri 240 . . . 4  |-  ( ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B )  <->  ( x  =  <. y ,  z
>.  /\  ( y  e.  A  /\  z  e.  B ) ) )
15142exbii 1574 . . 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 2825 . . . 4  |-  x  e. 
_V
1716, 4, 5brpprod3b 24812 . . 3  |-  ( xpprod (  _E  ,  _E  ) <. A ,  B >.  <->  E. y E. z ( x  =  <. y ,  z >.  /\  y  _E  A  /\  z  _E  B ) )
18 elxp 4743 . . 3  |-  ( x  e.  ( A  X.  B )  <->  E. y E. z ( x  = 
<. y ,  z >.  /\  ( y  e.  A  /\  z  e.  B
) ) )
1915, 17, 183bitr4ri 269 . 2  |-  ( x  e.  ( A  X.  B )  <->  xpprod (  _E  ,  _E  ) <. A ,  B >. )
201, 2, 3, 8, 19brtxpsd3 24821 1  |-  ( <. A ,  B >.Cart C  <-> 
C  =  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701   _Vcvv 2822   <.cop 3677   class class class wbr 4060    _E cep 4340    X. cxp 4724  pprodcpprod 24759  Cartccart 24769
This theorem is referenced by:  brimg  24861  brrestrict  24873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-eprel 4342  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-1st 6164  df-2nd 6165  df-symdif 24747  df-txp 24780  df-pprod 24781  df-cart 24791
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