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Theorem dmin 4902
Description: The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
dmin  |-  dom  ( A  i^i  B )  C_  ( dom  A  i^i  dom  B )

Proof of Theorem dmin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1599 . . 3  |-  ( E. y ( <. x ,  y >.  e.  A  /\  <. x ,  y
>.  e.  B )  -> 
( E. y <.
x ,  y >.  e.  A  /\  E. y <. x ,  y >.  e.  B ) )
2 vex 2804 . . . . 5  |-  x  e. 
_V
32eldm2 4893 . . . 4  |-  ( x  e.  dom  ( A  i^i  B )  <->  E. y <. x ,  y >.  e.  ( A  i^i  B
) )
4 elin 3371 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  i^i  B
)  <->  ( <. x ,  y >.  e.  A  /\  <. x ,  y
>.  e.  B ) )
54exbii 1572 . . . 4  |-  ( E. y <. x ,  y
>.  e.  ( A  i^i  B )  <->  E. y ( <.
x ,  y >.  e.  A  /\  <. x ,  y >.  e.  B
) )
63, 5bitri 240 . . 3  |-  ( x  e.  dom  ( A  i^i  B )  <->  E. y
( <. x ,  y
>.  e.  A  /\  <. x ,  y >.  e.  B
) )
7 elin 3371 . . . 4  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( x  e.  dom  A  /\  x  e.  dom  B ) )
82eldm2 4893 . . . . 5  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
92eldm2 4893 . . . . 5  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
108, 9anbi12i 678 . . . 4  |-  ( ( x  e.  dom  A  /\  x  e.  dom  B )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. y <. x ,  y >.  e.  B ) )
117, 10bitri 240 . . 3  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. y <. x ,  y
>.  e.  B ) )
121, 6, 113imtr4i 257 . 2  |-  ( x  e.  dom  ( A  i^i  B )  ->  x  e.  ( dom  A  i^i  dom  B )
)
1312ssriv 3197 1  |-  dom  ( A  i^i  B )  C_  ( dom  A  i^i  dom  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    e. wcel 1696    i^i cin 3164    C_ wss 3165   <.cop 3656   dom cdm 4705
This theorem is referenced by:  rnin  5106  psssdm2  14340  domintrefb  25166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715
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