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Theorem iunss2 3947
Description: A subclass condition on the members of two indexed classes 
C ( x ) and  D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 3858. (Contributed by NM, 9-Dec-2004.)
Assertion
Ref Expression
iunss2  |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  ->  U_ x  e.  A  C  C_  U_ y  e.  B  D )
Distinct variable groups:    x, y    x, B    y, C    x, D
Allowed substitution hints:    A( x, y)    B( y)    C( x)    D( y)

Proof of Theorem iunss2
StepHypRef Expression
1 ssiun 3944 . . 3  |-  ( E. y  e.  B  C  C_  D  ->  C  C_  U_ y  e.  B  D )
21ralimi 2618 . 2  |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  ->  A. x  e.  A  C  C_  U_ y  e.  B  D )
3 iunss 3943 . 2  |-  ( U_ x  e.  A  C  C_ 
U_ y  e.  B  D 
<-> 
A. x  e.  A  C  C_  U_ y  e.  B  D )
42, 3sylibr 203 1  |-  ( A. x  e.  A  E. y  e.  B  C  C_  D  ->  U_ x  e.  A  C  C_  U_ y  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wral 2543   E.wrex 2544    C_ wss 3152   U_ciun 3905
This theorem is referenced by:  iunxdif2  3950  oaass  6559  odi  6577  omass  6578  oelim2  6593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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