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Theorem dfiun3 5089
Description: Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfiun3.1  |-  B  e. 
_V
Assertion
Ref Expression
dfiun3  |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )

Proof of Theorem dfiun3
StepHypRef Expression
1 dfiun3g 5087 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
2 dfiun3.1 . . 3  |-  B  e. 
_V
32a1i 11 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2780 1  |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   _Vcvv 2967   U.cuni 4086   U_ciun 4166    e. cmpt 4345   ran crn 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-cnv 4843  df-dm 4845  df-rn 4846
This theorem is referenced by:  tgrest  18738  dstfrvunirn  26809  mblfinlem2  28382  volsupnfl  28389  comppfsc  28532  istotbnd3  28623  sstotbnd  28627
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