ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opabssxp Unicode version

Theorem opabssxp 4583
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 4169 . 2  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
3 df-xp 4515 . 2  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
42, 3sseqtrri 3102 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 1465    C_ wss 3041   {copab 3958    X. cxp 4507
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-in 3047  df-ss 3054  df-opab 3960  df-xp 4515
This theorem is referenced by:  brab2ga  4584  dmoprabss  5821  ecopovsym  6493  ecopovtrn  6494  ecopover  6495  ecopovsymg  6496  ecopovtrng  6497  ecopoverg  6498  enqex  7136  ltrelnq  7141  enq0ex  7215  ltrelpr  7281  enrex  7513  ltrelsr  7514  ltrelre  7609  ltrelxr  7793  dvdszrcl  11425  lmfval  12288
  Copyright terms: Public domain W3C validator