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Theorem dfiun3g 4766
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 3815 . 2  |-  ( A. x  e.  A  B  e.  C  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
2 eqid 2117 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 4757 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43unieqi 3716 . 2  |-  U. ran  ( x  e.  A  |->  B )  =  U. { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2168 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 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   U.cuni 3706   U_ciun 3783    |-> cmpt 3959   ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  dfiun3  4768  iunon  6149  tgiun  12169
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