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Theorem fnfvelrnd 7102
Description: A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
fnfvelrnd.1 (𝜑𝐹 Fn 𝐴)
fnfvelrnd.2 (𝜑𝐵𝐴)
Assertion
Ref Expression
fnfvelrnd (𝜑 → (𝐹𝐵) ∈ ran 𝐹)

Proof of Theorem fnfvelrnd
StepHypRef Expression
1 fnfvelrnd.1 . 2 (𝜑𝐹 Fn 𝐴)
2 fnfvelrnd.2 . 2 (𝜑𝐵𝐴)
3 fnfvelrn 7100 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)
41, 2, 3syl2anc 584 1 (𝜑 → (𝐹𝐵) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  ran crn 5690   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  ghmqusnsg  19313  ghmquskerlem3  19317  ghmqusker  19318  noseqrdglem  28326  ssmapsn  45159  limsupgtlem  45733
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