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Theorem iunss2 3949
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 3860. (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 3946 . . 3  |-  ( E. y  e.  B  C  C_  D  ->  C  C_  U_ y  e.  B  D )
21ralimi 2620 . 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 3945 . 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 205 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 6   A.wral 2545   E.wrex 2546    C_ wss 3154   U_ciun 3907
This theorem is referenced by:  iunxdif2  3952  oaass  6555  odi  6573  omass  6574  oelim2  6589
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rex 2551  df-v 2792  df-in 3161  df-ss 3168  df-iun 3909
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