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Theorem dmin 5220
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 1680 . . 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 3112 . . . . 5  |-  x  e. 
_V
32eldm2 5211 . . . 4  |-  ( x  e.  dom  ( A  i^i  B )  <->  E. y <. x ,  y >.  e.  ( A  i^i  B
) )
4 elin 3683 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  i^i  B
)  <->  ( <. x ,  y >.  e.  A  /\  <. x ,  y
>.  e.  B ) )
54exbii 1668 . . . 4  |-  ( E. y <. x ,  y
>.  e.  ( A  i^i  B )  <->  E. y ( <.
x ,  y >.  e.  A  /\  <. x ,  y >.  e.  B
) )
63, 5bitri 249 . . 3  |-  ( x  e.  dom  ( A  i^i  B )  <->  E. y
( <. x ,  y
>.  e.  A  /\  <. x ,  y >.  e.  B
) )
7 elin 3683 . . . 4  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( x  e.  dom  A  /\  x  e.  dom  B ) )
82eldm2 5211 . . . . 5  |-  ( x  e.  dom  A  <->  E. y <. x ,  y >.  e.  A )
92eldm2 5211 . . . . 5  |-  ( x  e.  dom  B  <->  E. y <. x ,  y >.  e.  B )
108, 9anbi12i 697 . . . 4  |-  ( ( x  e.  dom  A  /\  x  e.  dom  B )  <->  ( E. y <. x ,  y >.  e.  A  /\  E. y <. x ,  y >.  e.  B ) )
117, 10bitri 249 . . 3  |-  ( x  e.  ( dom  A  i^i  dom  B )  <->  ( E. y <. x ,  y
>.  e.  A  /\  E. y <. x ,  y
>.  e.  B ) )
121, 6, 113imtr4i 266 . 2  |-  ( x  e.  dom  ( A  i^i  B )  ->  x  e.  ( dom  A  i^i  dom  B )
)
1312ssriv 3503 1  |-  dom  ( A  i^i  B )  C_  ( dom  A  i^i  dom  B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   E.wex 1613    e. wcel 1819    i^i cin 3470    C_ wss 3471   <.cop 4038   dom cdm 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-dm 5018
This theorem is referenced by:  rnin  5422  psssdm2  15971  hauseqcn  28030
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