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Theorem relopabi 4635
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 3960 . . . 4  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
31, 2eqtri 2138 . . 3  |-  A  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
4 vex 2663 . . . . . . . 8  |-  x  e. 
_V
5 vex 2663 . . . . . . . 8  |-  y  e. 
_V
64, 5opelvv 4559 . . . . . . 7  |-  <. x ,  y >.  e.  ( _V  X.  _V )
7 eleq1 2180 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( _V  X.  _V ) 
<-> 
<. x ,  y >.  e.  ( _V  X.  _V ) ) )
86, 7mpbiri 167 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  z  e.  ( _V  X.  _V )
)
98adantr 274 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
109exlimivv 1852 . . . 4  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
1110abssi 3142 . . 3  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  C_  ( _V  X.  _V )
123, 11eqsstri 3099 . 2  |-  A  C_  ( _V  X.  _V )
13 df-rel 4516 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1412, 13mpbir 145 1  |-  Rel  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103   _Vcvv 2660    C_ wss 3041   <.cop 3500   {copab 3958    X. cxp 4507   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515  df-rel 4516
This theorem is referenced by:  relopab  4636  mptrel  4637  reli  4638  rele  4639  relcnv  4887  cotr  4890  relco  5007  reloprab  5787  reldmoprab  5824  eqer  6429  ecopover  6495  ecopoverg  6498  relen  6606  reldom  6607  enq0er  7211  aprcl  8375  climrel  11004  brstruct  11879
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