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Theorem dminss 5095
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 1718 . . . . . . 7  |-  ( ( x  e.  A  /\  x R y )  ->  E. x ( x  e.  A  /\  x R y ) )
21ancoms 439 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  E. x ( x  e.  A  /\  x R y ) )
3 vex 2791 . . . . . . 7  |-  y  e. 
_V
43elima2 5018 . . . . . 6  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
52, 4sylibr 203 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y  e.  ( R
" A ) )
6 simpl 443 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  x R y )
7 vex 2791 . . . . . . 7  |-  x  e. 
_V
83, 7brcnv 4864 . . . . . 6  |-  ( y `' R x  <->  x R
y )
96, 8sylibr 203 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y `' R x )
105, 9jca 518 . . . 4  |-  ( ( x R y  /\  x  e.  A )  ->  ( y  e.  ( R " A )  /\  y `' R x ) )
1110eximi 1563 . . 3  |-  ( E. y ( x R y  /\  x  e.  A )  ->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
127eldm 4876 . . . . 5  |-  ( x  e.  dom  R  <->  E. y  x R y )
1312anbi1i 676 . . . 4  |-  ( ( x  e.  dom  R  /\  x  e.  A
)  <->  ( E. y  x R y  /\  x  e.  A ) )
14 elin 3358 . . . 4  |-  ( x  e.  ( dom  R  i^i  A )  <->  ( x  e.  dom  R  /\  x  e.  A ) )
15 19.41v 1842 . . . 4  |-  ( E. y ( x R y  /\  x  e.  A )  <->  ( E. y  x R y  /\  x  e.  A )
)
1613, 14, 153bitr4i 268 . . 3  |-  ( x  e.  ( dom  R  i^i  A )  <->  E. y
( x R y  /\  x  e.  A
) )
177elima2 5018 . . 3  |-  ( x  e.  ( `' R " ( R " A
) )  <->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
1811, 16, 173imtr4i 257 . 2  |-  ( x  e.  ( dom  R  i^i  A )  ->  x  e.  ( `' R "
( R " A
) ) )
1918ssriv 3184 1  |-  ( dom 
R  i^i  A )  C_  ( `' R "
( R " A
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   "cima 4692
This theorem is referenced by:  lmhmlsp  15806  cnclsi  17001  kgencn3  17253  kqsat  17422  kqcldsat  17424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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