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Theorem relopabi 4992
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 4259 . . . 4  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
31, 2eqtri 2455 . . 3  |-  A  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
4 vex 2951 . . . . . . . 8  |-  x  e. 
_V
5 vex 2951 . . . . . . . 8  |-  y  e. 
_V
64, 5opelvv 4916 . . . . . . 7  |-  <. x ,  y >.  e.  ( _V  X.  _V )
7 eleq1 2495 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( _V  X.  _V ) 
<-> 
<. x ,  y >.  e.  ( _V  X.  _V ) ) )
86, 7mpbiri 225 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  z  e.  ( _V  X.  _V )
)
98adantr 452 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
109exlimivv 1645 . . . 4  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
1110abssi 3410 . . 3  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  C_  ( _V  X.  _V )
123, 11eqsstri 3370 . 2  |-  A  C_  ( _V  X.  _V )
13 df-rel 4877 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1412, 13mpbir 201 1  |-  Rel  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    C_ wss 3312   <.cop 3809   {copab 4257    X. cxp 4868   Rel wrel 4875
This theorem is referenced by:  relopab  4993  reli  4994  rele  4995  relcnv  5234  cotr  5238  relco  5360  reloprab  6114  reldmoprab  6150  relrpss  6515  eqer  6930  ecopover  7000  relen  7106  reldom  7107  relwdom  7526  fpwwe2lem2  8499  fpwwe2lem3  8500  fpwwe2lem6  8502  fpwwe2lem7  8503  fpwwe2lem9  8505  fpwwe2lem11  8507  fpwwe2lem12  8508  fpwwe2lem13  8509  fpwwelem  8512  fpwwe  8513  climrel  12278  rlimrel  12279  brstruct  13469  sscrel  14005  gaorber  15077  sylow2a  15245  efgrelexlema  15373  efgrelexlemb  15374  efgcpbllemb  15379  tpsexOLD  16976  2ndcctbss  17510  vitalilem1  19492  lgsquadlem1  21130  lgsquadlem2  21131  reluhgra  21328  relumgra  21341  reluslgra  21360  relusgra  21361  iscusgra0  21458  cusgrasize  21479  relrngo  21957  drngoi  21987  isdivrngo  22011  vcrel  22018  h2hlm  22475  hlimi  22682  relae  24583  mptrel  25384  fnerel  26338  refrel  26349  filnetlem3  26400  brabg2  26408  heiborlem3  26513  heiborlem4  26514  isdrngo1  26563  riscer  26595  prter1  26719  prter3  26722  rellindf  27246  frisusgrapr  28318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-rel 4877
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