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Theorem dfiun3g 4931
Description: Alternate definition of indexed union when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfiun3g  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )

Proof of Theorem dfiun3g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfiun2g 3935 . 2  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
2 eqid 2283 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 4925 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43unieqi 3837 . 2  |-  U. ran  ( x  e.  A  |->  B )  =  U. { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2333 1  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. ran  ( x  e.  A  |->  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   U.cuni 3827   U_ciun 3905    e. cmpt 4077   ran crn 4690
This theorem is referenced by:  dfiun3  4933  iunon  6355  onoviun  6360  gruiun  8421  tgiun  16717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700
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