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Theorem afvfundmfveq 27933
Description: If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvfundmfveq  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )

Proof of Theorem afvfundmfveq
StepHypRef Expression
1 dfafv2 27927 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iftrue 3737 . 2  |-  ( F defAt 
A  ->  if ( F defAt  A ,  ( F `
 A ) ,  _V )  =  ( F `  A ) )
31, 2syl5eq 2479 1  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   _Vcvv 2948   ifcif 3731   ` cfv 5446   defAt wdfat 27902  '''cafv 27903
This theorem is referenced by:  afvnufveq  27942  afvfvn0fveq  27945  afv0nbfvbi  27946  afveu  27948  fnbrafvb  27949  afvelrn  27963  afvres  27967  tz6.12-afv  27968  dmfcoafv  27970  afvco2  27971  rlimdmafv  27972  aovfundmoveq  27976
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-un 3317  df-if 3732  df-fv 5454  df-afv 27906
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