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Theorem iinss2 4351
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 3083 . . . . 5  |-  y  e. 
_V
2 eliin 4305 . . . . 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 2788 . . . 4  |-  ( A. x  e.  A  y  e.  B  ->  ( x  e.  A  ->  y  e.  B ) )
53, 4sylbi 198 . . 3  |-  ( y  e.  |^|_ x  e.  A  B  ->  ( x  e.  A  ->  y  e.  B ) )
65com12 32 . 2  |-  ( x  e.  A  ->  (
y  e.  |^|_ x  e.  A  B  ->  y  e.  B ) )
76ssrdv 3470 1  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    e. wcel 1872   A.wral 2771   _Vcvv 3080    C_ wss 3436   |^|_ciin 4300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-v 3082  df-in 3443  df-ss 3450  df-iin 4302
This theorem is referenced by:  dmiin  5097  gruiin  9242  txtube  20653
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