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Theorem funiunfvdm 5758
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 5757. (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 5757 . 2  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
2 imadmrn 4976 . . . 4  |-  ( F
" dom  F )  =  ran  F
3 fndm 5311 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
43imaeq2d 4966 . . . 4  |-  ( F  Fn  A  ->  ( F " dom  F )  =  ( F " A ) )
52, 4eqtr3id 2224 . . 3  |-  ( F  Fn  A  ->  ran  F  =  ( F " A ) )
65unieqd 3818 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. ( F
" A ) )
71, 6eqtrd 2210 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   U.cuni 3807   U_ciun 3884   dom cdm 4623   ran crn 4624   "cima 4626    Fn wfn 5207   ` cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220
This theorem is referenced by:  funiunfvdmf  5759  eluniimadm  5760
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