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Theorem iunss1 4343
Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunss1  |-  ( A 
C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  B  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iunss1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssrexv 3570 . . 3  |-  ( A 
C_  B  ->  ( E. x  e.  A  y  e.  C  ->  E. x  e.  B  y  e.  C ) )
2 eliun 4336 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
3 eliun 4336 . . 3  |-  ( y  e.  U_ x  e.  B  C  <->  E. x  e.  B  y  e.  C )
41, 2, 33imtr4g 270 . 2  |-  ( A 
C_  B  ->  (
y  e.  U_ x  e.  A  C  ->  y  e.  U_ x  e.  B  C ) )
54ssrdv 3515 1  |-  ( A 
C_  B  ->  U_ x  e.  A  C  C_  U_ x  e.  B  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   E.wrex 2818    C_ wss 3481   U_ciun 4331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-v 3120  df-in 3488  df-ss 3495  df-iun 4333
This theorem is referenced by:  iuneq1  4345  iunxdif2  4379  oelim2  7256  fsumiun  13615  ovolfiniun  21780  uniioovol  21856  usgreghash2spotv  24881  volsupnfl  29977  bnj1413  33526  bnj1408  33527
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