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Theorem dminss 5413
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 1801 . . . . . . 7  |-  ( ( x  e.  A  /\  x R y )  ->  E. x ( x  e.  A  /\  x R y ) )
21ancoms 453 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  E. x ( x  e.  A  /\  x R y ) )
3 vex 3111 . . . . . . 7  |-  y  e. 
_V
43elima2 5336 . . . . . 6  |-  ( y  e.  ( R " A )  <->  E. x
( x  e.  A  /\  x R y ) )
52, 4sylibr 212 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y  e.  ( R
" A ) )
6 simpl 457 . . . . . 6  |-  ( ( x R y  /\  x  e.  A )  ->  x R y )
7 vex 3111 . . . . . . 7  |-  x  e. 
_V
83, 7brcnv 5178 . . . . . 6  |-  ( y `' R x  <->  x R
y )
96, 8sylibr 212 . . . . 5  |-  ( ( x R y  /\  x  e.  A )  ->  y `' R x )
105, 9jca 532 . . . 4  |-  ( ( x R y  /\  x  e.  A )  ->  ( y  e.  ( R " A )  /\  y `' R x ) )
1110eximi 1630 . . 3  |-  ( E. y ( x R y  /\  x  e.  A )  ->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
127eldm 5193 . . . . 5  |-  ( x  e.  dom  R  <->  E. y  x R y )
1312anbi1i 695 . . . 4  |-  ( ( x  e.  dom  R  /\  x  e.  A
)  <->  ( E. y  x R y  /\  x  e.  A ) )
14 elin 3682 . . . 4  |-  ( x  e.  ( dom  R  i^i  A )  <->  ( x  e.  dom  R  /\  x  e.  A ) )
15 19.41v 1940 . . . 4  |-  ( E. y ( x R y  /\  x  e.  A )  <->  ( E. y  x R y  /\  x  e.  A )
)
1613, 14, 153bitr4i 277 . . 3  |-  ( x  e.  ( dom  R  i^i  A )  <->  E. y
( x R y  /\  x  e.  A
) )
177elima2 5336 . . 3  |-  ( x  e.  ( `' R " ( R " A
) )  <->  E. y
( y  e.  ( R " A )  /\  y `' R x ) )
1811, 16, 173imtr4i 266 . 2  |-  ( x  e.  ( dom  R  i^i  A )  ->  x  e.  ( `' R "
( R " A
) ) )
1918ssriv 3503 1  |-  ( dom 
R  i^i  A )  C_  ( `' R "
( R " A
) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   E.wex 1591    e. wcel 1762    i^i cin 3470    C_ wss 3471   class class class wbr 4442   `'ccnv 4993   dom cdm 4994   "cima 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007
This theorem is referenced by:  lmhmlsp  17473  cnclsi  19534  kgencn3  19789  kqsat  19962  kqcldsat  19964  cfilucfilOLD  20802  cfilucfil  20803
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