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Theorem dfatelrn 45516
Description: The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.)
Assertion
Ref Expression
dfatelrn (𝐹 defAt 𝐴 → (𝐹𝐴) ∈ ran 𝐹)

Proof of Theorem dfatelrn
StepHypRef Expression
1 df-dfat 45504 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 funressndmfvrn 45431 . . 3 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
32ancoms 459 . 2 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) ∈ ran 𝐹)
41, 3sylbi 216 1 (𝐹 defAt 𝐴 → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  {csn 4606  dom cdm 5653  ran crn 5654  cres 5655  Fun wfun 6510  cfv 6516   defAt wdfat 45501
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-fv 6524  df-dfat 45504
This theorem is referenced by: (None)
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