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Theorem iuniin 4253
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 2893 . . . 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 3025 . . . . . 6  |-  z  e. 
_V
3 eliin 4248 . . . . . 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 2866 . . . 4  |-  ( E. x  e.  A  z  e.  |^|_ y  e.  B  C 
<->  E. x  e.  A  A. y  e.  B  z  e.  C )
6 eliun 4247 . . . . 5  |-  ( z  e.  U_ x  e.  A  C  <->  E. x  e.  A  z  e.  C )
76ralbii 2796 . . . 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 269 . . 3  |-  ( E. x  e.  A  z  e.  |^|_ y  e.  B  C  ->  A. y  e.  B  z  e.  U_ x  e.  A  C )
9 eliun 4247 . . 3  |-  ( z  e.  U_ x  e.  A  |^|_ y  e.  B  C 
<->  E. x  e.  A  z  e.  |^|_ y  e.  B  C )
10 eliin 4248 . . . 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 269 . 2  |-  ( z  e.  U_ x  e.  A  |^|_ y  e.  B  C  ->  z  e.  |^|_ y  e.  B  U_ x  e.  A  C )
1312ssriv 3411 1  |-  U_ x  e.  A  |^|_ y  e.  B  C  C_  |^|_ y  e.  B  U_ x  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    e. wcel 1872   A.wral 2714   E.wrex 2715   _Vcvv 3022    C_ wss 3379   U_ciun 4242   |^|_ciin 4243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-v 3024  df-in 3386  df-ss 3393  df-iun 4244  df-iin 4245
This theorem is referenced by: (None)
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