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Theorem elrnrexdmb 5307
Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Assertion
Ref Expression
elrnrexdmb  |-  ( Fun 
F  ->  ( Y  e.  ran  F  <->  E. x  e.  dom  F  Y  =  ( F `  x
) ) )
Distinct variable groups:    x, F    x, Y

Proof of Theorem elrnrexdmb
StepHypRef Expression
1 funfn 4931 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fvelrnb 5221 . . 3  |-  ( F  Fn  dom  F  -> 
( Y  e.  ran  F  <->  E. x  e.  dom  F ( F `  x
)  =  Y ) )
31, 2sylbi 114 . 2  |-  ( Fun 
F  ->  ( Y  e.  ran  F  <->  E. x  e.  dom  F ( F `
 x )  =  Y ) )
4 eqcom 2042 . . 3  |-  ( Y  =  ( F `  x )  <->  ( F `  x )  =  Y )
54rexbii 2331 . 2  |-  ( E. x  e.  dom  F  Y  =  ( F `  x )  <->  E. x  e.  dom  F ( F `
 x )  =  Y )
63, 5syl6bbr 187 1  |-  ( Fun 
F  ->  ( Y  e.  ran  F  <->  E. x  e.  dom  F  Y  =  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    e. wcel 1393   E.wrex 2307   dom cdm 4345   ran crn 4346   Fun wfun 4896    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by: (None)
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