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Theorem funiunfvdm 5632
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 5631. (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 5631 . 2  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
2 imadmrn 4861 . . . 4  |-  ( F
" dom  F )  =  ran  F
3 fndm 5192 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
43imaeq2d 4851 . . . 4  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
52, 4syl5eqr 2164 . . 3  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
65unieqd 3717 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. ( F
" A ) )
71, 6eqtrd 2150 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   U.cuni 3706   U_ciun 3783   dom cdm 4509   ran crn 4510   "cima 4512    Fn wfn 5088   ` cfv 5093
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-sbc 2883  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-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  funiunfvdmf  5633  eluniimadm  5634
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