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Theorem afvfundmfveq 37591
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 37585 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iftrue 3891 . 2  |-  ( F defAt 
A  ->  if ( F defAt  A ,  ( F `
 A ) ,  _V )  =  ( F `  A ) )
31, 2syl5eq 2455 1  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   _Vcvv 3059   ifcif 3885   ` cfv 5569   defAt wdfat 37566  '''cafv 37567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rab 2763  df-v 3061  df-un 3419  df-if 3886  df-fv 5577  df-afv 37570
This theorem is referenced by:  afvnufveq  37600  afvfvn0fveq  37603  afv0nbfvbi  37604  afveu  37606  fnbrafvb  37607  afvelrn  37621  afvres  37625  tz6.12-afv  37626  dmfcoafv  37628  afvco2  37629  rlimdmafv  37630  aovfundmoveq  37634
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