Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndfatafv2 Structured version   Visualization version   GIF version

Theorem ndfatafv2 43633
Description: The alternate function value at a class 𝐴 if the function is not defined at this set 𝐴. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndfatafv2 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)

Proof of Theorem ndfatafv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-afv2 43631 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
2 iffalse 4459 . 2 𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = 𝒫 ran 𝐹)
31, 2syl5eq 2871 1 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  ifcif 4450  𝒫 cpw 4522   cuni 4825   class class class wbr 5053  ran crn 5544  cio 6301   defAt wdfat 43538  ''''cafv2 43630
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-if 4451  df-afv2 43631
This theorem is referenced by:  ndfatafv2undef  43634  ndfatafv2nrn  43643  afv2ndefb  43646  afv20defat  43654
  Copyright terms: Public domain W3C validator