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Theorem ecoptocl 6516
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 6481 . . 3  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  E. z  e.  ( B  X.  C
) A  =  [
z ] R )
2 eqid 2139 . . . . 5  |-  ( B  X.  C )  =  ( B  X.  C
)
3 eceq1 6464 . . . . . . 7  |-  ( <.
x ,  y >.  =  z  ->  [ <. x ,  y >. ] R  =  [ z ] R
)
43eqeq2d 2151 . . . . . 6  |-  ( <.
x ,  y >.  =  z  ->  ( A  =  [ <. x ,  y >. ] R  <->  A  =  [ z ] R ) )
54imbi1d 230 . . . . 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 2142 . . . . . 6  |-  ( A  =  [ <. x ,  y >. ] R  ->  ( ph  <->  ps )
)
96, 8syl5ibcom 154 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ( A  =  [ <. x ,  y >. ] R  ->  ps )
)
102, 5, 9optocl 4615 . . . 4  |-  ( z  e.  ( B  X.  C )  ->  ( A  =  [ z ] R  ->  ps )
)
1110rexlimiv 2543 . . 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 2234 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2417   <.cop 3530    X. cxp 4537   [cec 6427   /.cqs 6428
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435
This theorem is referenced by:  2ecoptocl  6517  3ecoptocl  6518  mulidnq  7204  recexnq  7205  ltsonq  7213  distrnq0  7274  addassnq0  7277  ltposr  7578  0idsr  7582  1idsr  7583  00sr  7584  recexgt0sr  7588  archsr  7597  srpospr  7598  map2psrprg  7620
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