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Theorem iuniin 3793
Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iuniin  |-  U_ x  e.  A  |^|_ y  e.  B  C  C_  |^|_ y  e.  B  U_ x  e.  A  C
Distinct variable groups:    x, y    y, A    x, B
Allowed substitution hints:    A( x)    B( y)    C( x, y)

Proof of Theorem iuniin
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 r19.12 2515 . . . 4  |-  ( E. x  e.  A  A. y  e.  B  z  e.  C  ->  A. y  e.  B  E. x  e.  A  z  e.  C )
2 vex 2663 . . . . . 6  |-  z  e. 
_V
3 eliin 3788 . . . . . 6  |-  ( z  e.  _V  ->  (
z  e.  |^|_ y  e.  B  C  <->  A. y  e.  B  z  e.  C ) )
42, 3ax-mp 5 . . . . 5  |-  ( z  e.  |^|_ y  e.  B  C 
<-> 
A. y  e.  B  z  e.  C )
54rexbii 2419 . . . 4  |-  ( E. x  e.  A  z  e.  |^|_ y  e.  B  C 
<->  E. x  e.  A  A. y  e.  B  z  e.  C )
6 eliun 3787 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
76ralbii 2418 . . . 4  |-  ( A. y  e.  B  z  e.  U_ x  e.  A  C 
<-> 
A. y  e.  B  E. x  e.  A  z  e.  C )
81, 5, 73imtr4i 200 . . 3  |-  ( E. x  e.  A  z  e.  |^|_ y  e.  B  C  ->  A. y  e.  B  z  e.  U_ x  e.  A  C )
9 eliun 3787 . . 3  |-  ( z  e.  U_ x  e.  A  |^|_ y  e.  B  C 
<->  E. x  e.  A  z  e.  |^|_ y  e.  B  C )
10 eliin 3788 . . . 4  |-  ( z  e.  _V  ->  (
z  e.  |^|_ y  e.  B  U_ x  e.  A  C  <->  A. y  e.  B  z  e.  U_ x  e.  A  C
) )
112, 10ax-mp 5 . . 3  |-  ( z  e.  |^|_ y  e.  B  U_ x  e.  A  C  <->  A. y  e.  B  z  e.  U_ x  e.  A  C )
128, 9, 113imtr4i 200 . 2  |-  ( z  e.  U_ x  e.  A  |^|_ y  e.  B  C  ->  z  e.  |^|_ y  e.  B  U_ x  e.  A  C )
1312ssriv 3071 1  |-  U_ x  e.  A  |^|_ y  e.  B  C  C_  |^|_ y  e.  B  U_ x  e.  A  C
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660    C_ wss 3041   U_ciun 3783   |^|_ciin 3784
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-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-in 3047  df-ss 3054  df-iun 3785  df-iin 3786
This theorem is referenced by: (None)
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