Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvelrn Unicode version

Theorem afvelrn 28030
Description: A function's value belongs to its range, analogous to fvelrn 5661. (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 5293 . . . . . 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 27974 . . . 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 28001 . . . 4  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
76eqcomd 2288 . . 3  |-  ( F defAt 
A  ->  ( F `  A )  =  ( F''' A ) )
85, 7syl 15 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  =  ( F''' A ) )
9 fvelrn 5661 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
108, 9eqeltrrd 2358 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 1623    e. wcel 1684   {csn 3640   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249   ` cfv 5255   defAt wdfat 27971  '''cafv 27972
This theorem is referenced by:  fnafvelrn  28031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-dfat 27974  df-afv 27975
  Copyright terms: Public domain W3C validator