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Theorem dmin 4886
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 )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem dmin
StepHypRef Expression
1 19.40 1597 . . 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 2793 . . . . 5  |-  x  e. 
_V
32eldm2 4877 . . . 4  |-  ( x  e.  dom  (  A  i^i  B )  <->  E. y <. x ,  y >.  e.  ( A  i^i  B
) )
4 elin 3360 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  i^i  B
)  <->  ( <. x ,  y >.  e.  A  /\  <. x ,  y
>.  e.  B ) )
54exbii 1570 . . . 4  |-  ( E. y <. x ,  y
>.  e.  ( A  i^i  B )  <->  E. y ( <.
x ,  y >.  e.  A  /\  <. x ,  y >.  e.  B
) )
63, 5bitri 242 . . 3  |-  ( x  e.  dom  (  A  i^i  B )  <->  E. y
( <. x ,  y
>.  e.  A  /\  <. x ,  y >.  e.  B
) )
7 elin 3360 . . . 4  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( x  e.  dom  A  /\  x  e.  dom  B ) )
82eldm2 4877 . . . . 5  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
92eldm2 4877 . . . . 5  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
108, 9anbi12i 680 . . . 4  |-  ( ( x  e.  dom  A  /\  x  e.  dom  B )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. y <. x ,  y >.  e.  B ) )
117, 10bitri 242 . . 3  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. y <. x ,  y
>.  e.  B ) )
121, 6, 113imtr4i 259 . 2  |-  ( x  e.  dom  (  A  i^i  B )  ->  x  e.  ( dom  A  i^i  dom  B )
)
1312ssriv 3186 1  |-  dom  (  A  i^i  B )  C_  ( dom  A  i^i  dom  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1529    e. wcel 1685    i^i cin 3153    C_ wss 3154   <.cop 3645   dom cdm 4689
This theorem is referenced by:  rnin  5090  psssdm2  14319  domintrefb  24462
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-dm 4699
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