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Theorem ecopqsi 6953
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 4902 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  -> 
<. B ,  C >.  e.  ( A  X.  A
) )
2 ecopqsi.1 . . . 4  |-  R  e. 
_V
32ecelqsi 6952 . . 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 2526 . 2  |-  ( <. B ,  C >.  e.  ( A  X.  A
)  ->  [ <. B ,  C >. ] R  e.  S )
61, 5syl 16 1  |-  ( ( B  e.  A  /\  C  e.  A )  ->  [ <. B ,  C >. ] R  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    X. cxp 4868   [cec 6895   /.cqs 6896
This theorem is referenced by:  brecop  6989  recexsrlem  8970
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-13 1727  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  ax-un 4693
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-eu 2284  df-mo 2285  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-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-ec 6899  df-qs 6903
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