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Theorem iinss 3953
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iinss  |-  ( E. x  e.  A  B  C_  C  ->  |^|_ x  e.  A  B  C_  C
)
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem iinss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . 4  |-  y  e. 
_V
2 eliin 3910 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
4 ssel 3174 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
54reximi 2650 . . . 4  |-  ( E. x  e.  A  B  C_  C  ->  E. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
6 r19.36av 2688 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  -> 
y  e.  C )  ->  ( A. x  e.  A  y  e.  B  ->  y  e.  C
) )
75, 6syl 15 . . 3  |-  ( E. x  e.  A  B  C_  C  ->  ( A. x  e.  A  y  e.  B  ->  y  e.  C ) )
83, 7syl5bi 208 . 2  |-  ( E. x  e.  A  B  C_  C  ->  ( y  e.  |^|_ x  e.  A  B  ->  y  e.  C
) )
98ssrdv 3185 1  |-  ( E. x  e.  A  B  C_  C  ->  |^|_ x  e.  A  B  C_  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   |^|_ciin 3906
This theorem is referenced by:  riinn0  3976  reliin  4807  cnviin  5212  iiner  6731  scott0  7556  cfslb  7892  ptbasfi  17276  iscmet3  18719  iintlem2  25611  fnemeet1  26315  pmapglb2N  29960  pmapglb2xN  29961
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-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iin 3908
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