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Theorem dminss 5083
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
StepHypRef Expression
1 19.8a 1758 . . . . . . 7  |-  ( ( x  e.  A  /\  x R y )  ->  E. x ( x  e.  A  /\  x R y ) )
21ancoms 441 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  E. x ( x  e.  A  /\  x R y ) )
3 vex 2766 . . . . . . 7  |-  y  e. 
_V
43elima2 5006 . . . . . 6  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
52, 4sylibr 205 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y  e.  ( R
" A ) )
6 simpl 445 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  x R y )
7 vex 2766 . . . . . . 7  |-  x  e. 
_V
83, 7brcnv 4852 . . . . . 6  |-  ( y `' R x  <->  x R
y )
96, 8sylibr 205 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y `' R x )
105, 9jca 520 . . . 4  |-  ( ( x R y  /\  x  e.  A )  ->  ( y  e.  ( R " A )  /\  y `' R x ) )
1110eximi 1574 . . 3  |-  ( E. y ( x R y  /\  x  e.  A )  ->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
127eldm 4864 . . . . 5  |-  ( x  e.  dom  R  <->  E. y  x R y )
1312anbi1i 679 . . . 4  |-  ( ( x  e.  dom  R  /\  x  e.  A
)  <->  ( E. y  x R y  /\  x  e.  A ) )
14 elin 3333 . . . 4  |-  ( x  e.  ( dom  R  i^i  A )  <->  ( x  e.  dom  R  /\  x  e.  A ) )
15 19.41v 2035 . . . 4  |-  ( E. y ( x R y  /\  x  e.  A )  <->  ( E. y  x R y  /\  x  e.  A )
)
1613, 14, 153bitr4i 270 . . 3  |-  ( x  e.  ( dom  R  i^i  A )  <->  E. y
( x R y  /\  x  e.  A
) )
177elima2 5006 . . 3  |-  ( x  e.  ( `' R " ( R " A
) )  <->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
1811, 16, 173imtr4i 259 . 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 360   E.wex 1537    e. wcel 1621    i^i cin 3126    C_ wss 3127   class class class wbr 3997   `'ccnv 4660   dom cdm 4661   "cima 4664
This theorem is referenced by:  lmhmlsp  15768  cnclsi  16963  kgencn3  17215  kqsat  17384  kqcldsat  17386
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682
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