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Theorem afvfundmfveq 27326
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 27320 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iftrue 3647 . 2  |-  ( F defAt 
A  ->  if ( F defAt  A ,  ( F `
 A ) ,  _V )  =  ( F `  A ) )
31, 2syl5eq 2402 1  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   _Vcvv 2864   ifcif 3641   ` cfv 5334   defAt wdfat 27294  '''cafv 27295
This theorem is referenced by:  afvnufveq  27335  afvfvn0fveq  27338  afv0nbfvbi  27339  afveu  27341  fnbrafvb  27342  afvelrn  27356  afvres  27360  tz6.12-afv  27361  dmfcoafv  27363  afvco2  27364  rlimdmafv  27365  aovfundmoveq  27369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-un 3233  df-if 3642  df-fv 5342  df-afv 27298
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