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Theorem elrnrexdm 6023
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 2468 . . . . . 6  |-  ( Y  e.  ran  F  ->  Y  =  Y )
21ancli 551 . . . . 5  |-  ( Y  e.  ran  F  -> 
( Y  e.  ran  F  /\  Y  =  Y ) )
32adantl 466 . . . 4  |-  ( ( Fun  F  /\  Y  e.  ran  F )  -> 
( Y  e.  ran  F  /\  Y  =  Y ) )
4 eqeq2 2482 . . . . 5  |-  ( y  =  Y  ->  ( Y  =  y  <->  Y  =  Y ) )
54rspcev 3214 . . . 4  |-  ( ( Y  e.  ran  F  /\  Y  =  Y
)  ->  E. y  e.  ran  F  Y  =  y )
63, 5syl 16 . . 3  |-  ( ( Fun  F  /\  Y  e.  ran  F )  ->  E. y  e.  ran  F  Y  =  y )
76ex 434 . 2  |-  ( Fun 
F  ->  ( Y  e.  ran  F  ->  E. y  e.  ran  F  Y  =  y ) )
8 funfn 5615 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
9 eqeq2 2482 . . . 4  |-  ( y  =  ( F `  x )  ->  ( Y  =  y  <->  Y  =  ( F `  x ) ) )
109rexrn 6021 . . 3  |-  ( F  Fn  dom  F  -> 
( E. y  e. 
ran  F  Y  =  y 
<->  E. x  e.  dom  F  Y  =  ( F `
 x ) ) )
118, 10sylbi 195 . 2  |-  ( Fun 
F  ->  ( E. y  e.  ran  F  Y  =  y  <->  E. x  e.  dom  F  Y  =  ( F `
 x ) ) )
127, 11sylibd 214 1  |-  ( Fun 
F  ->  ( Y  e.  ran  F  ->  E. x  e.  dom  F  Y  =  ( F `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   dom cdm 4999   ran crn 5000   Fun wfun 5580    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  nbgraf1olem1  24114  vdusgra0nedg  24581  usgravd0nedg  24591  bj-ccinftydisj  33688
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