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Theorem elrnrexdm 5655
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 2178 . . . . . 6  |-  ( Y  e.  ran  F  ->  Y  =  Y )
21ancli 323 . . . . 5  |-  ( Y  e.  ran  F  -> 
( Y  e.  ran  F  /\  Y  =  Y ) )
32adantl 277 . . . 4  |-  ( ( Fun  F  /\  Y  e.  ran  F )  -> 
( Y  e.  ran  F  /\  Y  =  Y ) )
4 eqeq2 2187 . . . . 5  |-  ( y  =  Y  ->  ( Y  =  y  <->  Y  =  Y ) )
54rspcev 2841 . . . 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 115 . 2  |-  ( Fun 
F  ->  ( Y  e.  ran  F  ->  E. y  e.  ran  F  Y  =  y ) )
8 funfn 5246 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
9 eqeq2 2187 . . . 4  |-  ( y  =  ( F `  x )  ->  ( Y  =  y  <->  Y  =  ( F `  x ) ) )
109rexrn 5653 . . 3  |-  ( F  Fn  dom  F  -> 
( E. y  e. 
ran  F  Y  =  y 
<->  E. x  e.  dom  F  Y  =  ( F `
 x ) ) )
118, 10sylbi 121 . 2  |-  ( Fun 
F  ->  ( E. y  e.  ran  F  Y  =  y  <->  E. x  e.  dom  F  Y  =  ( F `
 x ) ) )
127, 11sylibd 149 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   dom cdm 4626   ran crn 4627   Fun wfun 5210    Fn wfn 5211   ` cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224
This theorem is referenced by:  cc2lem  7264  ennnfonelemrnh  12411  ennnfonelemf1  12413  exmidsbthrlem  14652
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