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Theorem dfiunv2 4087
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 4055 . . . 4  |-  U_ y  e.  B  C  =  { w  |  E. y  e.  B  w  e.  C }
21a1i 11 . . 3  |-  ( x  e.  A  ->  U_ y  e.  B  C  =  { w  |  E. y  e.  B  w  e.  C } )
32iuneq2i 4071 . 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 4055 . 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 2919 . . . . 5  |-  z  e. 
_V
6 eleq1 2464 . . . . . 6  |-  ( w  =  z  ->  (
w  e.  C  <->  z  e.  C ) )
76rexbidv 2687 . . . . 5  |-  ( w  =  z  ->  ( E. y  e.  B  w  e.  C  <->  E. y  e.  B  z  e.  C ) )
85, 7elab 3042 . . . 4  |-  ( z  e.  { w  |  E. y  e.  B  w  e.  C }  <->  E. y  e.  B  z  e.  C )
98rexbii 2691 . . 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 2516 . 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 2428 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 1649    e. wcel 1721   {cab 2390   E.wrex 2667   U_ciun 4053
This theorem is referenced by:  2wot2wont  28083  2spot2iun2spont  28088  usg2spot2nb  28168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-v 2918  df-in 3287  df-ss 3294  df-iun 4055
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