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Theorem iinss 4324
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 3075 . . . 4  |-  y  e. 
_V
2 eliin 4279 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  y  e.  B ) )
31, 2ax-mp 5 . . 3  |-  ( y  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  y  e.  B )
4 ssel 3453 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
54reximi 2923 . . . 4  |-  ( E. x  e.  A  B  C_  C  ->  E. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
6 r19.36av 2968 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  -> 
y  e.  C )  ->  ( A. x  e.  A  y  e.  B  ->  y  e.  C
) )
75, 6syl 16 . . 3  |-  ( E. x  e.  A  B  C_  C  ->  ( A. x  e.  A  y  e.  B  ->  y  e.  C ) )
83, 7syl5bi 217 . 2  |-  ( E. x  e.  A  B  C_  C  ->  ( y  e.  |^|_ x  e.  A  B  ->  y  e.  C
) )
98ssrdv 3465 1  |-  ( E. x  e.  A  B  C_  C  ->  |^|_ x  e.  A  B  C_  C
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1758   A.wral 2796   E.wrex 2797   _Vcvv 3072    C_ wss 3431   |^|_ciin 4275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-v 3074  df-in 3438  df-ss 3445  df-iin 4277
This theorem is referenced by:  riinn0  4348  reliin  5064  cnviin  5477  iiner  7277  scott0  8199  cfslb  8541  ptbasfi  19281  iscmet3  20931  fnemeet1  28730  pmapglb2N  33734  pmapglb2xN  33735
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