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Theorem dfiun3g 4845
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 3883 . 2  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
2 eqid 2157 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 4836 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43unieqi 3784 . 2  |-  U. ran  ( x  e.  A  |->  B )  =  U. { y  |  E. x  e.  A  y  =  B }
51, 4eqtr4di 2208 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 1335    e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   U.cuni 3774   U_ciun 3851    |-> cmpt 4027   ran crn 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-cnv 4596  df-dm 4598  df-rn 4599
This theorem is referenced by:  dfiun3  4847  iunon  6233  tgiun  12543
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