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Theorem ecopqsi 6227
Description: "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)
Hypotheses
Ref Expression
ecopqsi.1  |-  R  e. 
_V
ecopqsi.2  |-  S  =  ( ( A  X.  A ) /. R
)
Assertion
Ref Expression
ecopqsi  |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S
)

Proof of Theorem ecopqsi
StepHypRef Expression
1 opelxpi 4402 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  -> 
<. B ,  C >.  e.  ( A  X.  A
) )
2 ecopqsi.1 . . . 4  |-  R  e. 
_V
32ecelqsi 6226 . . 3  |-  ( <. B ,  C >.  e.  ( A  X.  A
)  ->  [ <. B ,  C >. ] R  e.  ( ( A  X.  A ) /. R
) )
4 ecopqsi.2 . . 3  |-  S  =  ( ( A  X.  A ) /. R
)
53, 4syl6eleqr 2173 . 2  |-  ( <. B ,  C >.  e.  ( A  X.  A
)  ->  [ <. B ,  C >. ] R  e.  S )
61, 5syl 14 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   <.cop 3409    X. cxp 4369   [cec 6170   /.cqs 6171
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-ec 6174  df-qs 6178
This theorem is referenced by:  brecop  6262  recexgt0sr  7012
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