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Theorem iinss2 3954
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 2791 . . . . 5  |-  y  e. 
_V
2 eliin 3910 . . . . 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 2603 . . . 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 3185 1  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   |^|_ciin 3906
This theorem is referenced by:  dmiin  4922  gruiin  8432  txtube  17334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-in 3159  df-ss 3166  df-iin 3908
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