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Theorem afvfvn0fveq 41744
Description: If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvfvn0fveq ((𝐹𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹𝐴))

Proof of Theorem afvfvn0fveq
StepHypRef Expression
1 fvfundmfvn0 6367 . . 3 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 df-dfat 41710 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
31, 2sylibr 224 . 2 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
4 afvfundmfveq 41732 . 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 382   = wceq 1630  wcel 2144  wne 2942  c0 4061  {csn 4314  dom cdm 5249  cres 5251  Fun wfun 6025  cfv 6031   defAt wdfat 41707  '''cafv 41708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-dfat 41710  df-afv 41711
This theorem is referenced by:  aovovn0oveq  41788
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