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Theorem iunxdif2 3746
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 3743 . . 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 3108 . . . . 5  |-  ( A 
\  B )  C_  A
3 iunss1 3709 . . . . 5  |-  ( ( A  \  B ) 
C_  A  ->  U_ y  e.  ( A  \  B
) D  C_  U_ y  e.  A  D )
42, 3ax-mp 7 . . . 4  |-  U_ y  e.  ( A  \  B
) D  C_  U_ y  e.  A  D
5 iunxdif2.1 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65cbviunv 3737 . . . 4  |-  U_ x  e.  A  C  =  U_ y  e.  A  D
74, 6sseqtr4i 3041 . . 3  |-  U_ y  e.  ( A  \  B
) D  C_  U_ x  e.  A  C
81, 7jctil 305 . 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 3023 . 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 132 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 102    = wceq 1285   A.wral 2353   E.wrex 2354    \ cdif 2979    C_ wss 2982   U_ciun 3698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-iun 3700
This theorem is referenced by: (None)
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