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Theorem afvfundmfveq 27869
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 27863 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iftrue 3705 . 2  |-  ( F defAt 
A  ->  if ( F defAt  A ,  ( F `
 A ) ,  _V )  =  ( F `  A ) )
31, 2syl5eq 2448 1  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2916   ifcif 3699   ` cfv 5413   defAt wdfat 27838  '''cafv 27839
This theorem is referenced by:  afvnufveq  27878  afvfvn0fveq  27881  afv0nbfvbi  27882  afveu  27884  fnbrafvb  27885  afvelrn  27899  afvres  27903  tz6.12-afv  27904  dmfcoafv  27906  afvco2  27907  rlimdmafv  27908  aovfundmoveq  27912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-un 3285  df-if 3700  df-fv 5421  df-afv 27842
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