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Theorem iinss 3737
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 2605 . . . 4  |-  y  e. 
_V
2 eliin 3691 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 7 . . 3  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
4 ssel 2994 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
54reximi 2459 . . . 4  |-  ( E. x  e.  A  B  C_  C  ->  E. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
6 r19.36av 2506 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  -> 
y  e.  C )  ->  ( A. x  e.  A  y  e.  B  ->  y  e.  C
) )
75, 6syl 14 . . 3  |-  ( E. x  e.  A  B  C_  C  ->  ( A. x  e.  A  y  e.  B  ->  y  e.  C ) )
83, 7syl5bi 150 . 2  |-  ( E. x  e.  A  B  C_  C  ->  ( y  e.  |^|_ x  e.  A  B  ->  y  e.  C
) )
98ssrdv 3006 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 103    e. wcel 1434   A.wral 2349   E.wrex 2350   _Vcvv 2602    C_ wss 2974   |^|_ciin 3687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-iin 3689
This theorem is referenced by:  riinm  3758  reliin  4487  cnviinm  4889  iinerm  6244
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