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Theorem dminss 5277
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 1762 . . . . . . 7  |-  ( ( x  e.  A  /\  x R y )  ->  E. x ( x  e.  A  /\  x R y ) )
21ancoms 440 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  E. x ( x  e.  A  /\  x R y ) )
3 vex 2951 . . . . . . 7  |-  y  e. 
_V
43elima2 5200 . . . . . 6  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
52, 4sylibr 204 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y  e.  ( R
" A ) )
6 simpl 444 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  x R y )
7 vex 2951 . . . . . . 7  |-  x  e. 
_V
83, 7brcnv 5046 . . . . . 6  |-  ( y `' R x  <->  x R
y )
96, 8sylibr 204 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y `' R x )
105, 9jca 519 . . . 4  |-  ( ( x R y  /\  x  e.  A )  ->  ( y  e.  ( R " A )  /\  y `' R x ) )
1110eximi 1585 . . 3  |-  ( E. y ( x R y  /\  x  e.  A )  ->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
127eldm 5058 . . . . 5  |-  ( x  e.  dom  R  <->  E. y  x R y )
1312anbi1i 677 . . . 4  |-  ( ( x  e.  dom  R  /\  x  e.  A
)  <->  ( E. y  x R y  /\  x  e.  A ) )
14 elin 3522 . . . 4  |-  ( x  e.  ( dom  R  i^i  A )  <->  ( x  e.  dom  R  /\  x  e.  A ) )
15 19.41v 1924 . . . 4  |-  ( E. y ( x R y  /\  x  e.  A )  <->  ( E. y  x R y  /\  x  e.  A )
)
1613, 14, 153bitr4i 269 . . 3  |-  ( x  e.  ( dom  R  i^i  A )  <->  E. y
( x R y  /\  x  e.  A
) )
177elima2 5200 . . 3  |-  ( x  e.  ( `' R " ( R " A
) )  <->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
1811, 16, 173imtr4i 258 . 2  |-  ( x  e.  ( dom  R  i^i  A )  ->  x  e.  ( `' R "
( R " A
) ) )
1918ssriv 3344 1  |-  ( dom 
R  i^i  A )  C_  ( `' R "
( R " A
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    e. wcel 1725    i^i cin 3311    C_ wss 3312   class class class wbr 4204   `'ccnv 4868   dom cdm 4869   "cima 4872
This theorem is referenced by:  lmhmlsp  16113  cnclsi  17324  kgencn3  17578  kqsat  17751  kqcldsat  17753  cfilucfilOLD  18587  cfilucfil  18588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882
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