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Theorem dfiunv2 4305
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 4271 . . . 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 4288 . 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 4271 . 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 3034 . . . . 5  |-  z  e. 
_V
6 eleq1 2537 . . . . . 6  |-  ( w  =  z  ->  (
w  e.  C  <->  z  e.  C ) )
76rexbidv 2892 . . . . 5  |-  ( w  =  z  ->  ( E. y  e.  B  w  e.  C  <->  E. y  e.  B  z  e.  C ) )
85, 7elab 3173 . . . 4  |-  ( z  e.  { w  |  E. y  e.  B  w  e.  C }  <->  E. y  e.  B  z  e.  C )
98rexbii 2881 . . 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 2587 . 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 2497 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 setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   {cab 2457   E.wrex 2757   U_ciun 4269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-iun 4271
This theorem is referenced by:  2wot2wont  25693  2spot2iun2spont  25698  usg2spot2nb  25872
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