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Theorem afvelrn 38060
Description: A function's value belongs to its range, analogous to fvelrn 6030. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvelrn  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F''' A )  e.  ran  F )

Proof of Theorem afvelrn
StepHypRef Expression
1 funres 5640 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  { A } ) )
21anim1i 570 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( Fun  ( F  |` 
{ A } )  /\  A  e.  dom  F ) )
32ancomd 452 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
4 df-dfat 38008 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
53, 4sylibr 215 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  F defAt  A )
6 afvfundmfveq 38030 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
76eqcomd 2437 . . 3  |-  ( F defAt 
A  ->  ( F `  A )  =  ( F''' A ) )
85, 7syl 17 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  ( F''' A ) )
9 fvelrn 6030 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
108, 9eqeltrrd 2518 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F''' A )  e.  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {csn 4002   dom cdm 4854   ran crn 4855    |` cres 4856   Fun wfun 5595   ` cfv 5601   defAt wdfat 38005  '''cafv 38006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-dfat 38008  df-afv 38009
This theorem is referenced by:  fnafvelrn  38061
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