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Theorem iinss2 3970
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 2804 . . . . 5  |-  y  e. 
_V
2 eliin 3926 . . . . 5  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 8 . . . 4  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
4 rsp 2616 . . . 4  |-  ( A. x  e.  A  y  e.  B  ->  ( x  e.  A  ->  y  e.  B ) )
53, 4sylbi 187 . . 3  |-  ( y  e.  |^|_ x  e.  A  B  ->  ( x  e.  A  ->  y  e.  B ) )
65com12 27 . 2  |-  ( x  e.  A  ->  (
y  e.  |^|_ x  e.  A  B  ->  y  e.  B ) )
76ssrdv 3198 1  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   |^|_ciin 3922
This theorem is referenced by:  dmiin  4938  gruiin  8448  txtube  17350
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-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-iin 3924
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