Theorem afvfundmfveq 48411

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 48405	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		iftrue 4491	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = (𝐹‘𝐴))
3	1, 2	eqtrid 2810	1 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3454  ifcif 4485  ‘cfv 6555   defAt wdfat 48390  '''cafv 48391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-iota 6509  df-fun 6557  df-fv 6563  df-aiota 48358  df-dfat 48393  df-afv 48394

This theorem is used by:  afvnufveq  48420  afvfvn0fveq  48423  afv0nbfvbi  48424  afveu  48426  fnbrafvb  48427  afvelrn  48441  afvres  48445  tz6.12-afv  48446  dmfcoafv  48448  afvco2  48449  rlimdmafv  48450  aovfundmoveq  48454