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Theorem opabssxp 4434
Description: An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
Assertion
Ref Expression
opabssxp  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  ( A  X.  B )
Distinct variable groups:    x, y, A   
x, B, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem opabssxp
StepHypRef Expression
1 simpl 107 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  -> 
( x  e.  A  /\  y  e.  B
) )
21ssopab2i 4034 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
3 df-xp 4371 . 2  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
42, 3sseqtr4i 3033 1  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1434    C_ wss 2974   {copab 3840    X. cxp 4363
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-opab 3842  df-xp 4371
This theorem is referenced by:  brab2ga  4435  dmoprabss  5611  ecopovsym  6261  ecopovtrn  6262  ecopover  6263  ecopovsymg  6264  ecopovtrng  6265  ecopoverg  6266  enqex  6601  ltrelnq  6606  enq0ex  6680  ltrelpr  6746  enrex  6965  ltrelsr  6966  ltrelre  7052  ltrelxr  7229  dvdszrcl  10334
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