Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afv2fvn0fveq Structured version   Visualization version   GIF version

Theorem afv2fvn0fveq 47214
Description: If the function's value at an argument is not the empty set, it equals the alternate function value at this argument. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
afv2fvn0fveq ((𝐹𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹𝐴))

Proof of Theorem afv2fvn0fveq
StepHypRef Expression
1 fvfundmfvn0 6950 . . 3 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 df-dfat 47069 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
31, 2sylibr 234 . 2 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
4 dfatafv2eqfv 47211 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (𝐹𝐴))
53, 4syl 17 1 ((𝐹𝐴) ≠ ∅ → (𝐹''''𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  c0 4339  {csn 4631  dom cdm 5689  cres 5691  Fun wfun 6557  cfv 6563   defAt wdfat 47066  ''''cafv2 47158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-dfat 47069  df-afv2 47159
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator