Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfatafv2eqfv Structured version   Visualization version   GIF version

Theorem dfatafv2eqfv 44234
 Description: If a function is defined at a class 𝐴, the alternate function value equals the function's value at 𝐴. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
dfatafv2eqfv (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))

Proof of Theorem dfatafv2eqfv
StepHypRef Expression
1 dfafv22 44232 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹)
2 iftrue 4429 . 2 (𝐹 defAt 𝐴 → if(𝐹 defAt 𝐴, (𝐹𝐴), 𝒫 ran 𝐹) = (𝐹𝐴))
31, 2syl5eq 2805 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  ifcif 4423  𝒫 cpw 4497  ∪ cuni 4801  ran crn 5529  ‘cfv 6340   defAt wdfat 44089  ''''cafv2 44181 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-un 3865  df-if 4424  df-fv 6348  df-afv2 44182 This theorem is referenced by:  afv2rnfveq  44235  afv20fv0  44236  afv2fvn0fveq  44237
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