Theorem afvfundmfveq 44630

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 44624	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		iftrue 4465	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = (𝐹‘𝐴))
3	1, 2	eqtrid 2790	1 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1539  Vcvv 3432  ifcif 4459  ‘cfv 6433   defAt wdfat 44608  '''cafv 44609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-aiota 44577  df-dfat 44611  df-afv 44612

This theorem is referenced by:  afvnufveq  44639  afvfvn0fveq  44642  afv0nbfvbi  44643  afveu  44645  fnbrafvb  44646  afvelrn  44660  afvres  44664  tz6.12-afv  44665  dmfcoafv  44667  afvco2  44668  rlimdmafv  44669  aovfundmoveq  44673