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Theorem elrnrexdmb 5607
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 5200 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fvelrnb 5516 . . 3  |-  ( F  Fn  dom  F  -> 
( Y  e.  ran  F  <->  E. x  e.  dom  F ( F `  x
)  =  Y ) )
31, 2sylbi 120 . 2  |-  ( Fun 
F  ->  ( Y  e.  ran  F  <->  E. x  e.  dom  F ( F `
 x )  =  Y ) )
4 eqcom 2159 . . 3  |-  ( Y  =  ( F `  x )  <->  ( F `  x )  =  Y )
54rexbii 2464 . 2  |-  ( E. x  e.  dom  F  Y  =  ( F `  x )  <->  E. x  e.  dom  F ( F `
 x )  =  Y )
63, 5bitr4di 197 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 104    = wceq 1335    e. wcel 2128   E.wrex 2436   dom cdm 4586   ran crn 4587   Fun wfun 5164    Fn wfn 5165   ` cfv 5170
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178
This theorem is referenced by: (None)
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