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Theorem elrnrexdm 5932
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 2451 . . . . . 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 2464 . . . . 5  |-  ( y  =  Y  ->  ( Y  =  y  <->  Y  =  Y ) )
54rspcev 3155 . . . 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 5531 . . 3  |-  ( Fun 
F  <->  F  Fn  dom  F )
9 eqeq2 2464 . . . 4  |-  ( y  =  ( F `  x )  ->  ( Y  =  y  <->  Y  =  ( F `  x ) ) )
109rexrn 5930 . . 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 1370    e. wcel 1757   E.wrex 2793   dom cdm 4924   ran crn 4925   Fun wfun 5496    Fn wfn 5497   ` cfv 5502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-iota 5465  df-fun 5504  df-fn 5505  df-fv 5510
This theorem is referenced by:  nbgraf1olem1  23471  vdusgra0nedg  23699  usgravd0nedg  23703  bj-ccinftydisj  32818
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