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Theorem ss2iun 4108
Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ss2iun  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )

Proof of Theorem ss2iun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3342 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2781 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rexim 2810 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C )
)
42, 3syl 16 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E. x  e.  A  y  e.  B  ->  E. x  e.  A  y  e.  C ) )
5 eliun 4097 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
6 eliun 4097 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
74, 5, 63imtr4g 262 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( y  e.  U_ x  e.  A  B  ->  y  e.  U_ x  e.  A  C
) )
87ssrdv 3354 1  |-  ( A. x  e.  A  B  C_  C  ->  U_ x  e.  A  B  C_  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   U_ciun 4093
This theorem is referenced by:  iuneq2  4109  oawordri  6793  omwordri  6815  oewordri  6835  oeworde  6836  r1val1  7712  cfslb2n  8148  imasaddvallem  13754  dprdss  15587  tgcmp  17464  txcmplem1  17673  txcmplem2  17674  xkococnlem  17691  alexsubALT  18082  ptcmplem3  18085  metnrmlem2  18890  uniiccvol  19472  dvfval  19784  filnetlem3  26409  sstotbnd2  26483  equivtotbnd  26487  bnj1145  29362  bnj1136  29366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-iun 4095
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