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Theorem afvfundmfveq 45832
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) = (𝐹𝐴))

Proof of Theorem afvfundmfveq
StepHypRef Expression
1 dfafv2 45826 . 2 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
2 iftrue 4533 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), V) = (𝐹𝐴))
31, 2eqtrid 2784 1 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  Vcvv 3474  ifcif 4527  cfv 6540   defAt wdfat 45810  '''cafv 45811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-aiota 45779  df-dfat 45813  df-afv 45814
This theorem is referenced by:  afvnufveq  45841  afvfvn0fveq  45844  afv0nbfvbi  45845  afveu  45847  fnbrafvb  45848  afvelrn  45862  afvres  45866  tz6.12-afv  45867  dmfcoafv  45869  afvco2  45870  rlimdmafv  45871  aovfundmoveq  45875
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