Theorem afvfundmfveq 47143

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 47137	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		iftrue 4497	. 2 ⊢ (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = (𝐹‘𝐴))
3	1, 2	eqtrid 2777	1 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3450  ifcif 4491  ‘cfv 6514   defAt wdfat 47121  '''cafv 47122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-aiota 47090  df-dfat 47124  df-afv 47125

This theorem is referenced by:  afvnufveq  47152  afvfvn0fveq  47155  afv0nbfvbi  47156  afveu  47158  fnbrafvb  47159  afvelrn  47173  afvres  47177  tz6.12-afv  47178  dmfcoafv  47180  afvco2  47181  rlimdmafv  47182  aovfundmoveq  47186