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Theorem relopabi 5003
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 4270 . . . 4  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
31, 2eqtri 2458 . . 3  |-  A  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ph ) }
4 vex 2961 . . . . . . . 8  |-  x  e. 
_V
5 vex 2961 . . . . . . . 8  |-  y  e. 
_V
64, 5opelvv 4927 . . . . . . 7  |-  <. x ,  y >.  e.  ( _V  X.  _V )
7 eleq1 2498 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z  e.  ( _V  X.  _V ) 
<-> 
<. x ,  y >.  e.  ( _V  X.  _V ) ) )
86, 7mpbiri 226 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  z  e.  ( _V  X.  _V )
)
98adantr 453 . . . . 5  |-  ( ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
109exlimivv 1646 . . . 4  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( _V  X.  _V )
)
1110abssi 3420 . . 3  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  C_  ( _V  X.  _V )
123, 11eqsstri 3380 . 2  |-  A  C_  ( _V  X.  _V )
13 df-rel 4888 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
1412, 13mpbir 202 1  |-  Rel  A
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958    C_ wss 3322   <.cop 3819   {copab 4268    X. cxp 4879   Rel wrel 4886
This theorem is referenced by:  relopab  5004  reli  5005  rele  5006  relcnv  5245  cotr  5249  relco  5371  reloprab  6125  reldmoprab  6161  relrpss  6526  eqer  6941  ecopover  7011  relen  7117  reldom  7118  relwdom  7537  fpwwe2lem2  8512  fpwwe2lem3  8513  fpwwe2lem6  8515  fpwwe2lem7  8516  fpwwe2lem9  8518  fpwwe2lem11  8520  fpwwe2lem12  8521  fpwwe2lem13  8522  fpwwelem  8525  fpwwe  8526  climrel  12291  rlimrel  12292  brstruct  13482  sscrel  14018  gaorber  15090  sylow2a  15258  efgrelexlema  15386  efgrelexlemb  15387  efgcpbllemb  15392  tpsexOLD  16989  2ndcctbss  17523  vitalilem1  19505  lgsquadlem1  21143  lgsquadlem2  21144  reluhgra  21341  relumgra  21354  reluslgra  21373  relusgra  21374  iscusgra0  21471  cusgrasize  21492  relrngo  21970  drngoi  22000  isdivrngo  22024  vcrel  22031  h2hlm  22488  hlimi  22695  relae  24596  mptrel  25397  fnerel  26361  refrel  26372  filnetlem3  26423  brabg2  26431  heiborlem3  26536  heiborlem4  26537  isdrngo1  26586  riscer  26618  prter1  26742  prter3  26745  rellindf  27269  frisusgrapr  28455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887  df-rel 4888
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