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Theorem ecopqsi 6732
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 4737 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  -> 
<. B ,  C >.  e.  ( A  X.  A
) )
2 ecopqsi.1 . . . 4  |-  R  e. 
_V
32ecelqsi 6731 . . 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 2387 . 2  |-  ( <. B ,  C >.  e.  ( A  X.  A
)  ->  [ <. B ,  C >. ] R  e.  S )
61, 5syl 15 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703   [cec 6674   /.cqs 6675
This theorem is referenced by:  brecop  6767  recexsrlem  8741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678  df-qs 6682
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