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Theorem funiunfvdm 5671
Description: The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5670. (Contributed by Jim Kingdon, 10-Jan-2019.)
Assertion
Ref Expression
funiunfvdm  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem funiunfvdm
StepHypRef Expression
1 fniunfv 5670 . 2  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
2 imadmrn 4898 . . . 4  |-  ( F
" dom  F )  =  ran  F
3 fndm 5229 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
43imaeq2d 4888 . . . 4  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
52, 4syl5eqr 2187 . . 3  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
65unieqd 3754 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. ( F
" A ) )
71, 6eqtrd 2173 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   U.cuni 3743   U_ciun 3820   dom cdm 4546   ran crn 4547   "cima 4549    Fn wfn 5125   ` cfv 5130
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138
This theorem is referenced by:  funiunfvdmf  5672  eluniimadm  5673
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