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Theorem dfiunv2 26019
Description: Define double indexed union. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
dfiunv2  |-  U_ x  e.  A  U_ y  e.  B  C  =  {
z  |  E. x  e.  A  E. y  e.  B  z  e.  C }
Distinct variable groups:    x, z    y, z    z, A    z, B    z, C
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)

Proof of Theorem dfiunv2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 df-iun 3923 . . . 4  |-  U_ y  e.  B  C  =  { w  |  E. y  e.  B  w  e.  C }
21a1i 10 . . 3  |-  ( x  e.  A  ->  U_ y  e.  B  C  =  { w  |  E. y  e.  B  w  e.  C } )
32iuneq2i 3939 . 2  |-  U_ x  e.  A  U_ y  e.  B  C  =  U_ x  e.  A  {
w  |  E. y  e.  B  w  e.  C }
4 df-iun 3923 . 2  |-  U_ x  e.  A  { w  |  E. y  e.  B  w  e.  C }  =  { z  |  E. x  e.  A  z  e.  { w  |  E. y  e.  B  w  e.  C } }
5 vex 2804 . . . . 5  |-  z  e. 
_V
6 eleq1 2356 . . . . . 6  |-  ( w  =  z  ->  (
w  e.  C  <->  z  e.  C ) )
76rexbidv 2577 . . . . 5  |-  ( w  =  z  ->  ( E. y  e.  B  w  e.  C  <->  E. y  e.  B  z  e.  C ) )
85, 7elab 2927 . . . 4  |-  ( z  e.  { w  |  E. y  e.  B  w  e.  C }  <->  E. y  e.  B  z  e.  C )
98rexbii 2581 . . 3  |-  ( E. x  e.  A  z  e.  { w  |  E. y  e.  B  w  e.  C }  <->  E. x  e.  A  E. y  e.  B  z  e.  C )
109abbii 2408 . 2  |-  { z  |  E. x  e.  A  z  e.  {
w  |  E. y  e.  B  w  e.  C } }  =  {
z  |  E. x  e.  A  E. y  e.  B  z  e.  C }
113, 4, 103eqtri 2320 1  |-  U_ x  e.  A  U_ y  e.  B  C  =  {
z  |  E. x  e.  A  E. y  e.  B  z  e.  C }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   U_ciun 3921
This theorem is referenced by:  prismorcsetlemc  26020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-iun 3923
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