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Theorem iundif2 4360
Description: Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4347 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
Assertion
Ref Expression
iundif2  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Distinct variable group:    x, B
Allowed substitution hints:    A( x)    C( x)

Proof of Theorem iundif2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldif 3443 . . . . 5  |-  ( y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  C ) )
21rexbii 2925 . . . 4  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C ) )
3 r19.42v 2981 . . . 4  |-  ( E. x  e.  A  ( y  e.  B  /\  -.  y  e.  C
)  <->  ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C ) )
4 rexnal 2871 . . . . . 6  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  A. x  e.  A  y  e.  C )
5 vex 3081 . . . . . . 7  |-  y  e. 
_V
6 eliin 4299 . . . . . . 7  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
75, 6ax-mp 5 . . . . . 6  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
84, 7xchbinxr 312 . . . . 5  |-  ( E. x  e.  A  -.  y  e.  C  <->  -.  y  e.  |^|_ x  e.  A  C )
98anbi2i 698 . . . 4  |-  ( ( y  e.  B  /\  E. x  e.  A  -.  y  e.  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C )
)
102, 3, 93bitri 274 . . 3  |-  ( E. x  e.  A  y  e.  ( B  \  C )  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
11 eliun 4298 . . 3  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  E. x  e.  A  y  e.  ( B  \  C ) )
12 eldif 3443 . . 3  |-  ( y  e.  ( B  \  |^|_ x  e.  A  C
)  <->  ( y  e.  B  /\  -.  y  e.  |^|_ x  e.  A  C ) )
1310, 11, 123bitr4i 280 . 2  |-  ( y  e.  U_ x  e.  A  ( B  \  C )  <->  y  e.  ( B  \  |^|_ x  e.  A  C )
)
1413eqriv 2416 1  |-  U_ x  e.  A  ( B  \  C )  =  ( B  \  |^|_ x  e.  A  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   E.wrex 2774   _Vcvv 3078    \ cdif 3430   U_ciun 4293   |^|_ciin 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-v 3080  df-dif 3436  df-iun 4295  df-iin 4296
This theorem is referenced by:  iuncld  19984  pnrmopn  20283  alexsublem  20983  bcth3  22185  iundifdifd  28013  iundifdif  28014
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