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Theorem iunss1 4293
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 3528 . . 3  |-  ( A 
C_  B  ->  ( E. x  e.  A  y  e.  C  ->  E. x  e.  B  y  e.  C ) )
2 eliun 4286 . . 3  |-  ( y  e.  U_ x  e.  A  C  <->  E. x  e.  A  y  e.  C )
3 eliun 4286 . . 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 3473 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 1758   E.wrex 2800    C_ wss 3439   U_ciun 4282
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 1955  ax-ext 2432
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 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-in 3446  df-ss 3453  df-iun 4284
This theorem is referenced by:  iuneq1  4295  iunxdif2  4329  oelim2  7147  fsumiun  13405  ovolfiniun  21119  uniioovol  21195  volsupnfl  28604  usgreghash2spotv  30827  bnj1413  32378  bnj1408  32379
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