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Theorem eldmrexrnb 6014
Description: For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 5578 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 5578 of the value of a function,  ( F `  Y )  =  (/) may mean that the value of  F at  Y is the empty set or that  F is not defined at  Y. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Assertion
Ref Expression
eldmrexrnb  |-  ( ( Fun  F  /\  (/)  e/  ran  F )  ->  ( Y  e.  dom  F  <->  E. x  e.  ran  F  x  =  ( F `  Y
) ) )
Distinct variable groups:    x, F    x, Y

Proof of Theorem eldmrexrnb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eldmrexrn 6013 . . 3  |-  ( Fun 
F  ->  ( Y  e.  dom  F  ->  E. x  e.  ran  F  x  =  ( F `  Y
) ) )
21adantr 463 . 2  |-  ( ( Fun  F  /\  (/)  e/  ran  F )  ->  ( Y  e.  dom  F  ->  E. x  e.  ran  F  x  =  ( F `  Y
) ) )
3 eleq1 2526 . . . . 5  |-  ( x  =  ( F `  Y )  ->  (
x  e.  ran  F  <->  ( F `  Y )  e.  ran  F ) )
4 elnelne2 2802 . . . . . . . . 9  |-  ( ( ( F `  Y
)  e.  ran  F  /\  (/)  e/  ran  F
)  ->  ( F `  Y )  =/=  (/) )
5 n0 3793 . . . . . . . . . 10  |-  ( ( F `  Y )  =/=  (/)  <->  E. y  y  e.  ( F `  Y
) )
6 elfvdm 5874 . . . . . . . . . . 11  |-  ( y  e.  ( F `  Y )  ->  Y  e.  dom  F )
76exlimiv 1727 . . . . . . . . . 10  |-  ( E. y  y  e.  ( F `  Y )  ->  Y  e.  dom  F )
85, 7sylbi 195 . . . . . . . . 9  |-  ( ( F `  Y )  =/=  (/)  ->  Y  e.  dom  F )
94, 8syl 16 . . . . . . . 8  |-  ( ( ( F `  Y
)  e.  ran  F  /\  (/)  e/  ran  F
)  ->  Y  e.  dom  F )
109expcom 433 . . . . . . 7  |-  ( (/)  e/ 
ran  F  ->  ( ( F `  Y )  e.  ran  F  ->  Y  e.  dom  F ) )
1110adantl 464 . . . . . 6  |-  ( ( Fun  F  /\  (/)  e/  ran  F )  ->  ( ( F `  Y )  e.  ran  F  ->  Y  e.  dom  F ) )
1211com12 31 . . . . 5  |-  ( ( F `  Y )  e.  ran  F  -> 
( ( Fun  F  /\  (/)  e/  ran  F
)  ->  Y  e.  dom  F ) )
133, 12syl6bi 228 . . . 4  |-  ( x  =  ( F `  Y )  ->  (
x  e.  ran  F  ->  ( ( Fun  F  /\  (/)  e/  ran  F
)  ->  Y  e.  dom  F ) ) )
1413com13 80 . . 3  |-  ( ( Fun  F  /\  (/)  e/  ran  F )  ->  ( x  e.  ran  F  ->  (
x  =  ( F `
 Y )  ->  Y  e.  dom  F ) ) )
1514rexlimdv 2944 . 2  |-  ( ( Fun  F  /\  (/)  e/  ran  F )  ->  ( E. x  e.  ran  F  x  =  ( F `  Y )  ->  Y  e.  dom  F ) )
162, 15impbid 191 1  |-  ( ( Fun  F  /\  (/)  e/  ran  F )  ->  ( Y  e.  dom  F  <->  E. x  e.  ran  F  x  =  ( F `  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649    e/ wnel 2650   E.wrex 2805   (/)c0 3783   dom cdm 4988   ran crn 4989   Fun wfun 5564   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by: (None)
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