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

Theorem elrnrexdm 5527
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 2118 . . . . . 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 2127 . . . . 5  |-  ( y  =  Y  ->  ( Y  =  y  <->  Y  =  Y ) )
54rspcev 2763 . . . 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 5123 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
9 eqeq2 2127 . . . 4  |-  ( y  =  ( F `  x )  ->  ( Y  =  y  <->  Y  =  ( F `  x ) ) )
109rexrn 5525 . . 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 1316    e. wcel 1465   E.wrex 2394   dom cdm 4509   ran crn 4510   Fun wfun 5087    Fn wfn 5088   ` cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  ennnfonelemrnh  11856  ennnfonelemf1  11858  exmidsbthrlem  13144
  Copyright terms: Public domain W3C validator