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Theorem elrnrexdmb 5526
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 5121 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fvelrnb 5435 . . 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 2117 . . 3  |-  ( Y  =  ( F `  x )  <->  ( F `  x )  =  Y )
54rexbii 2417 . 2  |-  ( E. x  e.  dom  F  Y  =  ( F `  x )  <->  E. x  e.  dom  F ( F `
 x )  =  Y )
63, 5syl6bbr 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 1314    e. wcel 1463   E.wrex 2392   dom cdm 4507   ran crn 4508   Fun wfun 5085    Fn wfn 5086   ` cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by: (None)
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