Theorem afvnfundmuv 46332

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 46325	. 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V)
2		iffalse 4529	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = V)
3	1, 2	eqtrid 2776	1 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533  Vcvv 3466  ifcif 4520  ‘cfv 6533   defAt wdfat 46309  '''cafv 46310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-res 5678  df-iota 6485  df-fun 6535  df-fv 6541  df-aiota 46278  df-dfat 46312  df-afv 46313

This theorem is referenced by:  ndmafv  46333  nfunsnafv  46335  afvnufveq  46340  afvres  46365  afvco2  46369  aovnfundmuv  46375