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Theorem iunxdif2 3950
Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
Hypothesis
Ref Expression
iunxdif2.1  |-  ( x  =  y  ->  C  =  D )
Assertion
Ref Expression
iunxdif2  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) C  C_  D  ->  U_ y  e.  ( A  \  B ) D  =  U_ x  e.  A  C )
Distinct variable groups:    x, y, A    x, B, y    y, C    x, D
Allowed substitution hints:    C( x)    D( y)

Proof of Theorem iunxdif2
StepHypRef Expression
1 iunss2 3947 . . 3  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) C  C_  D  ->  U_ x  e.  A  C  C_  U_ y  e.  ( A  \  B
) D )
2 difss 3303 . . . . 5  |-  ( A 
\  B )  C_  A
3 iunss1 3916 . . . . 5  |-  ( ( A  \  B ) 
C_  A  ->  U_ y  e.  ( A  \  B
) D  C_  U_ y  e.  A  D )
42, 3ax-mp 8 . . . 4  |-  U_ y  e.  ( A  \  B
) D  C_  U_ y  e.  A  D
5 iunxdif2.1 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65cbviunv 3941 . . . 4  |-  U_ x  e.  A  C  =  U_ y  e.  A  D
74, 6sseqtr4i 3211 . . 3  |-  U_ y  e.  ( A  \  B
) D  C_  U_ x  e.  A  C
81, 7jctil 523 . 2  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) C  C_  D  ->  ( U_ y  e.  ( A  \  B
) D  C_  U_ x  e.  A  C  /\  U_ x  e.  A  C  C_ 
U_ y  e.  ( A  \  B ) D ) )
9 eqss 3194 . 2  |-  ( U_ y  e.  ( A  \  B ) D  = 
U_ x  e.  A  C 
<->  ( U_ y  e.  ( A  \  B
) D  C_  U_ x  e.  A  C  /\  U_ x  e.  A  C  C_ 
U_ y  e.  ( A  \  B ) D ) )
108, 9sylibr 203 1  |-  ( A. x  e.  A  E. y  e.  ( A  \  B ) C  C_  D  ->  U_ y  e.  ( A  \  B ) D  =  U_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152   U_ciun 3905
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-dif 3155  df-in 3159  df-ss 3166  df-iun 3907
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