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Theorem afvfundmfveq 31690
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 31684 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iftrue 3945 . 2  |-  ( F defAt 
A  ->  if ( F defAt  A ,  ( F `
 A ) ,  _V )  =  ( F `  A ) )
31, 2syl5eq 2520 1  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   _Vcvv 3113   ifcif 3939   ` cfv 5586   defAt wdfat 31665  '''cafv 31666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-un 3481  df-if 3940  df-fv 5594  df-afv 31669
This theorem is referenced by:  afvnufveq  31699  afvfvn0fveq  31702  afv0nbfvbi  31703  afveu  31705  fnbrafvb  31706  afvelrn  31720  afvres  31724  tz6.12-afv  31725  dmfcoafv  31727  afvco2  31728  rlimdmafv  31729  aovfundmoveq  31733
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