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Theorem afv2fvn0fveq 47276
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 6949 . . 3 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 df-dfat 47131 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
31, 2sylibr 234 . 2 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
4 dfatafv2eqfv 47273 . 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 1540  wcel 2108  wne 2940  c0 4333  {csn 4626  dom cdm 5685  cres 5687  Fun wfun 6555  cfv 6561   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-dfat 47131  df-afv2 47221
This theorem is referenced by: (None)
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