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Theorem dmin 5068
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 1619 . . 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 2951 . . . . 5  |-  x  e. 
_V
32eldm2 5059 . . . 4  |-  ( x  e.  dom  ( A  i^i  B )  <->  E. y <. x ,  y >.  e.  ( A  i^i  B
) )
4 elin 3522 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  i^i  B
)  <->  ( <. x ,  y >.  e.  A  /\  <. x ,  y
>.  e.  B ) )
54exbii 1592 . . . 4  |-  ( E. y <. x ,  y
>.  e.  ( A  i^i  B )  <->  E. y ( <.
x ,  y >.  e.  A  /\  <. x ,  y >.  e.  B
) )
63, 5bitri 241 . . 3  |-  ( x  e.  dom  ( A  i^i  B )  <->  E. y
( <. x ,  y
>.  e.  A  /\  <. x ,  y >.  e.  B
) )
7 elin 3522 . . . 4  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( x  e.  dom  A  /\  x  e.  dom  B ) )
82eldm2 5059 . . . . 5  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
92eldm2 5059 . . . . 5  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
108, 9anbi12i 679 . . . 4  |-  ( ( x  e.  dom  A  /\  x  e.  dom  B )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. y <. x ,  y >.  e.  B ) )
117, 10bitri 241 . . 3  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. y <. x ,  y
>.  e.  B ) )
121, 6, 113imtr4i 258 . 2  |-  ( x  e.  dom  ( A  i^i  B )  ->  x  e.  ( dom  A  i^i  dom  B )
)
1312ssriv 3344 1  |-  dom  ( A  i^i  B )  C_  ( dom  A  i^i  dom  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    e. wcel 1725    i^i cin 3311    C_ wss 3312   <.cop 3809   dom cdm 4869
This theorem is referenced by:  rnin  5272  psssdm2  14635  hauseqcn  24281
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  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-dm 4879
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