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Theorem ecopqsi 6347
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 4469 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  -> 
<. B ,  C >.  e.  ( A  X.  A
) )
2 ecopqsi.1 . . . 4  |-  R  e. 
_V
32ecelqsi 6346 . . 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 2181 . 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 1289    e. wcel 1438   _Vcvv 2619   <.cop 3449    X. cxp 4436   [cec 6290   /.cqs 6291
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-ec 6294  df-qs 6298
This theorem is referenced by:  brecop  6382  recexgt0sr  7319
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