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Theorem iinss 3776
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 2622 . . . 4  |-  y  e. 
_V
2 eliin 3730 . . . 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 3017 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
54reximi 2470 . . . 4  |-  ( E. x  e.  A  B  C_  C  ->  E. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
6 r19.36av 2518 . . . 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 3029 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 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619    C_ wss 2997   |^|_ciin 3726
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3003  df-ss 3010  df-iin 3728
This theorem is referenced by:  riinm  3797  reliin  4547  cnviinm  4959  iinerm  6344
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