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Theorem ecoptocl 6379
Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)
Hypotheses
Ref Expression
ecoptocl.1  |-  S  =  ( ( B  X.  C ) /. R
)
ecoptocl.2  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
ecoptocl.3  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ph )
Assertion
Ref Expression
ecoptocl  |-  ( A  e.  S  ->  ps )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, R, y    ps, x, y
Allowed substitution hints:    ph( x, y)    S( x, y)

Proof of Theorem ecoptocl
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elqsi 6344 . . 3  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  E. z  e.  ( B  X.  C
) A  =  [
z ] R )
2 eqid 2088 . . . . 5  |-  ( B  X.  C )  =  ( B  X.  C
)
3 eceq1 6327 . . . . . . 7  |-  ( <.
x ,  y >.  =  z  ->  [ <. x ,  y >. ] R  =  [ z ] R
)
43eqeq2d 2099 . . . . . 6  |-  ( <.
x ,  y >.  =  z  ->  ( A  =  [ <. x ,  y >. ] R  <->  A  =  [ z ] R ) )
54imbi1d 229 . . . . 5  |-  ( <.
x ,  y >.  =  z  ->  ( ( A  =  [ <. x ,  y >. ] R  ->  ps )  <->  ( A  =  [ z ] R  ->  ps ) ) )
6 ecoptocl.3 . . . . . 6  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ph )
7 ecoptocl.2 . . . . . . 7  |-  ( [
<. x ,  y >. ] R  =  A  ->  ( ph  <->  ps )
)
87eqcoms 2091 . . . . . 6  |-  ( A  =  [ <. x ,  y >. ] R  ->  ( ph  <->  ps )
)
96, 8syl5ibcom 153 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ( A  =  [ <. x ,  y >. ] R  ->  ps )
)
102, 5, 9optocl 4514 . . . 4  |-  ( z  e.  ( B  X.  C )  ->  ( A  =  [ z ] R  ->  ps )
)
1110rexlimiv 2483 . . 3  |-  ( E. z  e.  ( B  X.  C ) A  =  [ z ] R  ->  ps )
121, 11syl 14 . 2  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  ps )
13 ecoptocl.1 . 2  |-  S  =  ( ( B  X.  C ) /. R
)
1412, 13eleq2s 2182 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360   <.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-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
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  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-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:  2ecoptocl  6380  3ecoptocl  6381  mulidnq  6948  recexnq  6949  ltsonq  6957  distrnq0  7018  addassnq0  7021  ltposr  7309  0idsr  7313  1idsr  7314  00sr  7315  recexgt0sr  7319  archsr  7327  srpospr  7328
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