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Theorem iinss2 3865
Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
iinss2  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)

Proof of Theorem iinss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2689 . . . . 5  |-  y  e. 
_V
2 eliin 3818 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 5 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
4 rsp 2480 . . . 4  |-  ( A. x  e.  A  y  e.  B  ->  ( x  e.  A  ->  y  e.  B ) )
53, 4sylbi 120 . . 3  |-  ( y  e.  |^|_ x  e.  A  B  ->  ( x  e.  A  ->  y  e.  B ) )
65com12 30 . 2  |-  ( x  e.  A  ->  (
y  e.  |^|_ x  e.  A  B  ->  y  e.  B ) )
76ssrdv 3103 1  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1480   A.wral 2416   _Vcvv 2686    C_ wss 3071   |^|_ciin 3814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-iin 3816
This theorem is referenced by:  dmiin  4785
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