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Theorem iinss2 4107
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 2923 . . . . 5  |-  y  e. 
_V
2 eliin 4062 . . . . 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 2730 . . . 4  |-  ( A. x  e.  A  y  e.  B  ->  ( x  e.  A  ->  y  e.  B ) )
53, 4sylbi 188 . . 3  |-  ( y  e.  |^|_ x  e.  A  B  ->  ( x  e.  A  ->  y  e.  B ) )
65com12 29 . 2  |-  ( x  e.  A  ->  (
y  e.  |^|_ x  e.  A  B  ->  y  e.  B ) )
76ssrdv 3318 1  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   A.wral 2670   _Vcvv 2920    C_ wss 3284   |^|_ciin 4058
This theorem is referenced by:  dmiin  5076  gruiin  8645  txtube  17629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-v 2922  df-in 3291  df-ss 3298  df-iin 4060
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