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Theorem relopabi 4764
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
Hypothesis
Ref Expression
relopabi.1  |-  A  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
relopabi  |-  Rel  A

Proof of Theorem relopabi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 relopabi.1 . . . 4  |-  A  =  { <. x ,  y
>.  |  ph }
2 df-opab 4077 . . . 4  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
31, 2eqtri 2208 . . 3  |-  A  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
4 vex 2752 . . . . . . . 8  |-  x  e. 
_V
5 vex 2752 . . . . . . . 8  |-  y  e. 
_V
64, 5opelvv 4688 . . . . . . 7  |-  <. x ,  y >.  e.  ( _V  X.  _V )
7 eleq1 2250 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( _V  X.  _V ) 
<-> 
<. x ,  y >.  e.  ( _V  X.  _V ) ) )
86, 7mpbiri 168 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  z  e.  ( _V  X.  _V )
)
98adantr 276 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
109exlimivv 1906 . . . 4  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
1110abssi 3242 . . 3  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  C_  ( _V  X.  _V )
123, 11eqsstri 3199 . 2  |-  A  C_  ( _V  X.  _V )
13 df-rel 4645 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1412, 13mpbir 146 1  |-  Rel  A
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1363   E.wex 1502    e. wcel 2158   {cab 2173   _Vcvv 2749    C_ wss 3141   <.cop 3607   {copab 4075    X. cxp 4636   Rel wrel 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-opab 4077  df-xp 4644  df-rel 4645
This theorem is referenced by:  relopab  4765  mptrel  4767  reli  4768  rele  4769  relcnv  5018  cotr  5022  relco  5139  reloprab  5936  reldmoprab  5973  eqer  6580  ecopover  6646  ecopoverg  6649  relen  6757  reldom  6758  enq0er  7447  aprcl  8616  aptap  8620  climrel  11301  brstruct  12484
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