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Theorem ecoptocl 6995
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 6959 . . 3  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  E. z  e.  ( B  X.  C
) A  =  [
z ] R )
2 eqid 2437 . . . . 5  |-  ( B  X.  C )  =  ( B  X.  C
)
3 eceq1 6942 . . . . . . 7  |-  ( <.
x ,  y >.  =  z  ->  [ <. x ,  y >. ] R  =  [ z ] R
)
43eqeq2d 2448 . . . . . 6  |-  ( <.
x ,  y >.  =  z  ->  ( A  =  [ <. x ,  y >. ] R  <->  A  =  [ z ] R ) )
54imbi1d 310 . . . . 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 2440 . . . . . 6  |-  ( A  =  [ <. x ,  y >. ] R  ->  ( ph  <->  ps )
)
96, 8syl5ibcom 213 . . . . 5  |-  ( ( x  e.  B  /\  y  e.  C )  ->  ( A  =  [ <. x ,  y >. ] R  ->  ps )
)
102, 5, 9optocl 4953 . . . 4  |-  ( z  e.  ( B  X.  C )  ->  ( A  =  [ z ] R  ->  ps )
)
1110rexlimiv 2825 . . 3  |-  ( E. z  e.  ( B  X.  C ) A  =  [ z ] R  ->  ps )
121, 11syl 16 . 2  |-  ( A  e.  ( ( B  X.  C ) /. R )  ->  ps )
13 ecoptocl.1 . 2  |-  S  =  ( ( B  X.  C ) /. R
)
1412, 13eleq2s 2529 1  |-  ( A  e.  S  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707   <.cop 3818    X. cxp 4877   [cec 6904   /.cqs 6905
This theorem is referenced by:  2ecoptocl  6996  3ecoptocl  6997  0idsr  8973  1idsr  8974  00sr  8975  recexsrlem  8979  map2psrpr  8986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-ec 6908  df-qs 6912
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