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Theorem iuniin 4095
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 2811 . . . 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 2951 . . . . . 6  |-  z  e. 
_V
3 eliin 4090 . . . . . 6  |-  ( z  e.  _V  ->  (
z  e.  |^|_ y  e.  B  C  <->  A. y  e.  B  z  e.  C ) )
42, 3ax-mp 8 . . . . 5  |-  ( z  e.  |^|_ y  e.  B  C 
<-> 
A. y  e.  B  z  e.  C )
54rexbii 2722 . . . 4  |-  ( E. x  e.  A  z  e.  |^|_ y  e.  B  C 
<->  E. x  e.  A  A. y  e.  B  z  e.  C )
6 eliun 4089 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
76ralbii 2721 . . . 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 258 . . 3  |-  ( E. x  e.  A  z  e.  |^|_ y  e.  B  C  ->  A. y  e.  B  z  e.  U_ x  e.  A  C )
9 eliun 4089 . . 3  |-  ( z  e.  U_ x  e.  A  |^|_ y  e.  B  C 
<->  E. x  e.  A  z  e.  |^|_ y  e.  B  C )
10 eliin 4090 . . . 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 8 . . 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 258 . 2  |-  ( z  e.  U_ x  e.  A  |^|_ y  e.  B  C  ->  z  e.  |^|_ y  e.  B  U_ x  e.  A  C )
1312ssriv 3344 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 177    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   U_ciun 4085   |^|_ciin 4086
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-in 3319  df-ss 3326  df-iun 4087  df-iin 4088
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