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Theorem ecoptocl 6791
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 6755 . . 3  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  E. z  e.  ( B  X.  C
) A  =  [
z ] R )
2 eqid 2316 . . . . 5  |-  ( B  X.  C )  =  ( B  X.  C
)
3 eceq1 6738 . . . . . . 7  |-  ( <.
x ,  y >.  =  z  ->  [ <. x ,  y >. ] R  =  [ z ] R
)
43eqeq2d 2327 . . . . . 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 2319 . . . . . 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 4801 . . . 4  |-  ( z  e.  ( B  X.  C )  ->  ( A  =  [ z ] R  ->  ps )
)
1110rexlimiv 2695 . . 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 2408 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578   <.cop 3677    X. cxp 4724   [cec 6700   /.cqs 6701
This theorem is referenced by:  2ecoptocl  6792  3ecoptocl  6793  0idsr  8764  1idsr  8765  00sr  8766  recexsrlem  8770  map2psrpr  8777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-ec 6704  df-qs 6708
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