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Theorem dfiun3 4949
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 4947 . 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 10 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2625 1  |-  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801   U.cuni 3843   U_ciun 3921    e. cmpt 4093   ran crn 4706
This theorem is referenced by:  tgrest  16906  dstfrvunirn  23690  comppfsc  26410  istotbnd3  26598  sstotbnd  26602
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716
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