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Theorem afvelrn 28041
Description: A function's value belongs to its range, analogous to fvelrn 5663. (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 5295 . . . . . 6  |-  ( Fun 
F  ->  Fun  ( F  |`  { A } ) )
21anim1i 551 . . . . 5  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( Fun  ( F  |` 
{ A } )  /\  A  e.  dom  F ) )
32ancomd 438 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
4 df-dfat 27985 . . . 4  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
53, 4sylibr 203 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  F defAt  A )
6 afvfundmfveq 28012 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
76eqcomd 2290 . . 3  |-  ( F defAt 
A  ->  ( F `  A )  =  ( F''' A ) )
85, 7syl 15 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  ( F''' A ) )
9 fvelrn 5663 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
108, 9eqeltrrd 2360 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F''' A )  e.  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   {csn 3642   dom cdm 4691   ran crn 4692    |` cres 4693   Fun wfun 5251   ` cfv 5257   defAt wdfat 27982  '''cafv 27983
This theorem is referenced by:  fnafvelrn  28042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265  df-dfat 27985  df-afv 27986
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