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Theorem iinss 3922
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 2733 . . . 4  |-  y  e. 
_V
2 eliin 3876 . . . 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 3141 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
54reximi 2567 . . . 4  |-  ( E. x  e.  A  B  C_  C  ->  E. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
6 r19.36av 2621 . . . 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 151 . 2  |-  ( E. x  e.  A  B  C_  C  ->  ( y  e.  |^|_ x  e.  A  B  ->  y  e.  C
) )
98ssrdv 3153 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 104    e. wcel 2141   A.wral 2448   E.wrex 2449   _Vcvv 2730    C_ wss 3121   |^|_ciin 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-iin 3874
This theorem is referenced by:  riinm  3943  reliin  4731  cnviinm  5150  iinerm  6581
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