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Theorem ecoptocl 6748
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 6713 . . 3  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  E. z  e.  ( B  X.  C
) A  =  [
z ] R )
2 eqid 2283 . . . . 5  |-  ( B  X.  C )  =  ( B  X.  C
)
3 eceq1 6696 . . . . . . 7  |-  ( <.
x ,  y >.  =  z  ->  [ <. x ,  y >. ] R  =  [ z ] R
)
43eqeq2d 2294 . . . . . 6  |-  ( <.
x ,  y >.  =  z  ->  ( A  =  [ <. x ,  y >. ] R  <->  A  =  [ z ] R ) )
54imbi1d 308 . . . . 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 2286 . . . . . 6  |-  ( A  =  [ <. x ,  y >. ] R  ->  ( ph  <->  ps )
)
96, 8syl5ibcom 211 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ( A  =  [ <. x ,  y >. ] R  ->  ps )
)
102, 5, 9optocl 4764 . . . 4  |-  ( z  e.  ( B  X.  C )  ->  ( A  =  [ z ] R  ->  ps )
)
1110rexlimiv 2661 . . 3  |-  ( E. z  e.  ( B  X.  C ) A  =  [ z ] R  ->  ps )
121, 11syl 15 . 2  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  ps )
13 ecoptocl.1 . 2  |-  S  =  ( ( B  X.  C ) /. R
)
1412, 13eleq2s 2375 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   <.cop 3643    X. cxp 4687   [cec 6658   /.cqs 6659
This theorem is referenced by:  2ecoptocl  6749  3ecoptocl  6750  0idsr  8719  1idsr  8720  00sr  8721  recexsrlem  8725  map2psrpr  8732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-qs 6666
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