ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elrnrexdm Unicode version

Theorem elrnrexdm 5559
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, 8-Dec-2017.)
Assertion
Ref Expression
elrnrexdm  |-  ( Fun 
F  ->  ( Y  e.  ran  F  ->  E. x  e.  dom  F  Y  =  ( F `  x
) ) )
Distinct variable groups:    x, F    x, Y

Proof of Theorem elrnrexdm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eqidd 2140 . . . . . 6  |-  ( Y  e.  ran  F  ->  Y  =  Y )
21ancli 321 . . . . 5  |-  ( Y  e.  ran  F  -> 
( Y  e.  ran  F  /\  Y  =  Y ) )
32adantl 275 . . . 4  |-  ( ( Fun  F  /\  Y  e.  ran  F )  -> 
( Y  e.  ran  F  /\  Y  =  Y ) )
4 eqeq2 2149 . . . . 5  |-  ( y  =  Y  ->  ( Y  =  y  <->  Y  =  Y ) )
54rspcev 2789 . . . 4  |-  ( ( Y  e.  ran  F  /\  Y  =  Y
)  ->  E. y  e.  ran  F  Y  =  y )
63, 5syl 14 . . 3  |-  ( ( Fun  F  /\  Y  e.  ran  F )  ->  E. y  e.  ran  F  Y  =  y )
76ex 114 . 2  |-  ( Fun 
F  ->  ( Y  e.  ran  F  ->  E. y  e.  ran  F  Y  =  y ) )
8 funfn 5153 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
9 eqeq2 2149 . . . 4  |-  ( y  =  ( F `  x )  ->  ( Y  =  y  <->  Y  =  ( F `  x ) ) )
109rexrn 5557 . . 3  |-  ( F  Fn  dom  F  -> 
( E. y  e. 
ran  F  Y  =  y 
<->  E. x  e.  dom  F  Y  =  ( F `
 x ) ) )
118, 10sylbi 120 . 2  |-  ( Fun 
F  ->  ( E. y  e.  ran  F  Y  =  y  <->  E. x  e.  dom  F  Y  =  ( F `
 x ) ) )
127, 11sylibd 148 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    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E.wrex 2417   dom cdm 4539   ran crn 4540   Fun wfun 5117    Fn wfn 5118   ` cfv 5123
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  ennnfonelemrnh  11929  ennnfonelemf1  11931  exmidsbthrlem  13217
  Copyright terms: Public domain W3C validator