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

Theorem dminss 5043
Description: An upper bound for intersection with a domain. Theorem 40 of [Suppes] p. 66, who calls it "somewhat surprising". (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
dminss  |-  ( dom 
R  i^i  A )  C_  ( `' R "
( R " A
) )

Proof of Theorem dminss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.8a 1590 . . . . . . 7  |-  ( ( x  e.  A  /\  x R y )  ->  E. x ( x  e.  A  /\  x R y ) )
21ancoms 268 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  E. x ( x  e.  A  /\  x R y ) )
3 vex 2740 . . . . . . 7  |-  y  e. 
_V
43elima2 4976 . . . . . 6  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
52, 4sylibr 134 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y  e.  ( R
" A ) )
6 simpl 109 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  x R y )
7 vex 2740 . . . . . . 7  |-  x  e. 
_V
83, 7brcnv 4810 . . . . . 6  |-  ( y `' R x  <->  x R
y )
96, 8sylibr 134 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y `' R x )
105, 9jca 306 . . . 4  |-  ( ( x R y  /\  x  e.  A )  ->  ( y  e.  ( R " A )  /\  y `' R x ) )
1110eximi 1600 . . 3  |-  ( E. y ( x R y  /\  x  e.  A )  ->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
127eldm 4824 . . . . 5  |-  ( x  e.  dom  R  <->  E. y  x R y )
1312anbi1i 458 . . . 4  |-  ( ( x  e.  dom  R  /\  x  e.  A
)  <->  ( E. y  x R y  /\  x  e.  A ) )
14 elin 3318 . . . 4  |-  ( x  e.  ( dom  R  i^i  A )  <->  ( x  e.  dom  R  /\  x  e.  A ) )
15 19.41v 1902 . . . 4  |-  ( E. y ( x R y  /\  x  e.  A )  <->  ( E. y  x R y  /\  x  e.  A )
)
1613, 14, 153bitr4i 212 . . 3  |-  ( x  e.  ( dom  R  i^i  A )  <->  E. y
( x R y  /\  x  e.  A
) )
177elima2 4976 . . 3  |-  ( x  e.  ( `' R " ( R " A
) )  <->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
1811, 16, 173imtr4i 201 . 2  |-  ( x  e.  ( dom  R  i^i  A )  ->  x  e.  ( `' R "
( R " A
) ) )
1918ssriv 3159 1  |-  ( dom 
R  i^i  A )  C_  ( `' R "
( R " A
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104   E.wex 1492    e. wcel 2148    i^i cin 3128    C_ wss 3129   class class class wbr 4003   `'ccnv 4625   dom cdm 4626   "cima 4629
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-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator