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Theorem eluniimadm 5618
Description: Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
Assertion
Ref Expression
eluniimadm  |-  ( F  Fn  A  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
Distinct variable groups:    x, A    x, B    x, F

Proof of Theorem eluniimadm
StepHypRef Expression
1 eliun 3781 . 2  |-  ( B  e.  U_ x  e.  A  ( F `  x )  <->  E. x  e.  A  B  e.  ( F `  x ) )
2 funiunfvdm 5616 . . 3  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
32eleq2d 2182 . 2  |-  ( F  Fn  A  ->  ( B  e.  U_ x  e.  A  ( F `  x )  <->  B  e.  U. ( F " A
) ) )
41, 3syl5rbbr 194 1  |-  ( F  Fn  A  ->  ( B  e.  U. ( F " A )  <->  E. x  e.  A  B  e.  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1461   E.wrex 2389   U.cuni 3700   U_ciun 3777   "cima 4500    Fn wfn 5074   ` cfv 5079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-fv 5087
This theorem is referenced by: (None)
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