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Theorem ecopqsi 6568
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 4643 . 2  |-  ( ( B  e.  A  /\  C  e.  A )  -> 
<. B ,  C >.  e.  ( A  X.  A
) )
2 ecopqsi.1 . . . 4  |-  R  e. 
_V
32ecelqsi 6567 . . 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, 4eleqtrrdi 2264 . 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 103    = wceq 1348    e. wcel 2141   _Vcvv 2730   <.cop 3586    X. cxp 4609   [cec 6511   /.cqs 6512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515  df-qs 6519
This theorem is referenced by:  brecop  6603  recexgt0sr  7735  ltpsrprg  7765
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