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Theorem opabssxp 4699
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 109 . . 3  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ph )  -> 
( x  e.  A  /\  y  e.  B
) )
21ssopab2i 4276 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
3 df-xp 4631 . 2  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
42, 3sseqtrri 3190 1  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    e. wcel 2148    C_ wss 3129   {copab 4062    X. cxp 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3135  df-ss 3142  df-opab 4064  df-xp 4631
This theorem is referenced by:  brab2ga  4700  dmoprabss  5954  ecopovsym  6628  ecopovtrn  6629  ecopover  6630  ecopovsymg  6631  ecopovtrng  6632  ecopoverg  6633  netap  7250  2omotaplemap  7253  2omotaplemst  7254  enqex  7356  ltrelnq  7361  enq0ex  7435  ltrelpr  7501  enrex  7733  ltrelsr  7734  ltrelre  7829  ltrelxr  8014  dvdszrcl  11792  releqgg  13011  aprval  13271  aprap  13275  lmfval  13563
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