Theorem afvfundmfveq 47088

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

Step	Hyp	Ref	Expression
1		dfafv2 47082	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		iftrue 4537	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = (𝐹‘𝐴))
3	1, 2	eqtrid 2787	1 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3478  ifcif 4531  ‘cfv 6563   defAt wdfat 47066  '''cafv 47067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070

This theorem is referenced by:  afvnufveq  47097  afvfvn0fveq  47100  afv0nbfvbi  47101  afveu  47103  fnbrafvb  47104  afvelrn  47118  afvres  47122  tz6.12-afv  47123  dmfcoafv  47125  afvco2  47126  rlimdmafv  47127  aovfundmoveq  47131