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Theorem iunxdif2 4131
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 4128 . . 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 3466 . . . . 5  |-  ( A 
\  B )  C_  A
3 iunss1 4096 . . . . 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 4122 . . . 4  |-  U_ x  e.  A  C  =  U_ y  e.  A  D
74, 6sseqtr4i 3373 . . 3  |-  U_ y  e.  ( A  \  B
) D  C_  U_ x  e.  A  C
81, 7jctil 524 . 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 3355 . 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 204 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 359    = wceq 1652   A.wral 2697   E.wrex 2698    \ cdif 3309    C_ wss 3312   U_ciun 4085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-iun 4087
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