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Theorem iunxdif2 3831
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 3828 . . 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 3172 . . . . 5  |-  ( A 
\  B )  C_  A
3 iunss1 3794 . . . . 5  |-  ( ( A  \  B ) 
C_  A  ->  U_ y  e.  ( A  \  B
) D  C_  U_ y  e.  A  D )
42, 3ax-mp 5 . . . 4  |-  U_ y  e.  ( A  \  B
) D  C_  U_ y  e.  A  D
5 iunxdif2.1 . . . . 5  |-  ( x  =  y  ->  C  =  D )
65cbviunv 3822 . . . 4  |-  U_ x  e.  A  C  =  U_ y  e.  A  D
74, 6sseqtrri 3102 . . 3  |-  U_ y  e.  ( A  \  B
) D  C_  U_ x  e.  A  C
81, 7jctil 310 . 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 3082 . 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 133 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 103    = wceq 1316   A.wral 2393   E.wrex 2394    \ cdif 3038    C_ wss 3041   U_ciun 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-iun 3785
This theorem is referenced by: (None)
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