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Theorem afv2fvn0fveq 42113
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 6451 . . 3 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 df-dfat 41968 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
31, 2sylibr 226 . 2 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
4 dfatafv2eqfv 42110 . 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 385   = wceq 1653  wcel 2157  wne 2972  c0 4116  {csn 4369  dom cdm 5313  cres 5315  Fun wfun 6096  cfv 6102   defAt wdfat 41965  ''''cafv2 42057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-res 5325  df-iota 6065  df-fun 6104  df-fv 6110  df-dfat 41968  df-afv2 42058
This theorem is referenced by: (None)
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