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Theorem afv2fvn0fveq 43746
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 6699 . . 3 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 df-dfat 43601 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
31, 2sylibr 237 . 2 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
4 dfatafv2eqfv 43743 . 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 399   = wceq 1538  wcel 2115  wne 3014  c0 4276  {csn 4550  dom cdm 5542  cres 5544  Fun wfun 6337  cfv 6343   defAt wdfat 43598  ''''cafv2 43690
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-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-dfat 43601  df-afv2 43691
This theorem is referenced by: (None)
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