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Theorem funressndmfvrn 43478
Description: The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐴 is in the domain of the function 𝐹. It is sufficient that 𝐹 is a function at 𝐴. (Contributed by AV, 1-Sep-2022.)
Assertion
Ref Expression
funressndmfvrn ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)

Proof of Theorem funressndmfvrn
StepHypRef Expression
1 simpr 488 . 2 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom 𝐹)
2 fvressn 6905 . . . 4 (𝐴 ∈ dom 𝐹 → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
32adantl 485 . . 3 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) = (𝐹𝐴))
4 eldmressnsn 5876 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹 ↾ {𝐴}))
5 fvelrn 6825 . . . 4 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom (𝐹 ↾ {𝐴})) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
64, 5sylan2 595 . . 3 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 ↾ {𝐴})‘𝐴) ∈ ran (𝐹 ↾ {𝐴}))
73, 6eqeltrrd 2917 . 2 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran (𝐹 ↾ {𝐴}))
8 fvrnressn 6904 . 2 (𝐴 ∈ dom 𝐹 → ((𝐹𝐴) ∈ ran (𝐹 ↾ {𝐴}) → (𝐹𝐴) ∈ ran 𝐹))
91, 7, 8sylc 65 1 ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {csn 4548  dom cdm 5536  ran crn 5537  cres 5538  Fun wfun 6330  cfv 6336
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 5184  ax-nul 5191  ax-pr 5311
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-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344
This theorem is referenced by:  dfatelrn  43529
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