Theorem afvnfundmuv 45833

Description: If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
afvnfundmuv	⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Proof of Theorem afvnfundmuv

Step	Hyp	Ref	Expression
1		dfafv2 45826	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		iffalse 4536	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V)
3	1, 2	eqtrid 2784	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Syntax hints:  ¬ wn 3   → 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:  ndmafv  45834  nfunsnafv  45836  afvnufveq  45841  afvres  45866  afvco2  45870  aovnfundmuv  45876