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Theorem afvelrn 27999
Description: A function's value belongs to its range, analogous to fvelrn 5858. (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 5484 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  { A } ) )
21anim1i 552 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( Fun  ( F  |` 
{ A } )  /\  A  e.  dom  F ) )
32ancomd 439 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
4 df-dfat 27941 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
53, 4sylibr 204 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  F defAt  A )
6 afvfundmfveq 27969 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
76eqcomd 2440 . . 3  |-  ( F defAt 
A  ->  ( F `  A )  =  ( F''' A ) )
85, 7syl 16 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  ( F''' A ) )
9 fvelrn 5858 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
108, 9eqeltrrd 2510 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F''' A )  e.  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   dom cdm 4870   ran crn 4871    |` cres 4872   Fun wfun 5440   ` cfv 5446   defAt wdfat 27938  '''cafv 27939
This theorem is referenced by:  fnafvelrn  28000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-dfat 27941  df-afv 27942
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