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Theorem opabssxp 4613
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 108 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  -> 
( x  e.  A  /\  y  e.  B
) )
21ssopab2i 4199 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
3 df-xp 4545 . 2  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
42, 3sseqtrri 3132 1  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1480    C_ wss 3071   {copab 3988    X. cxp 4537
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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545
This theorem is referenced by:  brab2ga  4614  dmoprabss  5853  ecopovsym  6525  ecopovtrn  6526  ecopover  6527  ecopovsymg  6528  ecopovtrng  6529  ecopoverg  6530  enqex  7168  ltrelnq  7173  enq0ex  7247  ltrelpr  7313  enrex  7545  ltrelsr  7546  ltrelre  7641  ltrelxr  7825  dvdszrcl  11498  lmfval  12361
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