Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvfundmfveq Structured version   Unicode version

Theorem afvfundmfveq 38396
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 38390 . 2  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
2 iftrue 3916 . 2  |-  ( F defAt 
A  ->  if ( F defAt  A ,  ( F `
 A ) ,  _V )  =  ( F `  A ) )
31, 2syl5eq 2476 1  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438   _Vcvv 3082   ifcif 3910   ` cfv 5599   defAt wdfat 38371  '''cafv 38372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rab 2785  df-v 3084  df-un 3442  df-if 3911  df-fv 5607  df-afv 38375
This theorem is referenced by:  afvnufveq  38405  afvfvn0fveq  38408  afv0nbfvbi  38409  afveu  38411  fnbrafvb  38412  afvelrn  38426  afvres  38430  tz6.12-afv  38431  dmfcoafv  38433  afvco2  38434  rlimdmafv  38435  aovfundmoveq  38439
  Copyright terms: Public domain W3C validator