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Theorem fnrnafv 47112
Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6968. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fnrnafv (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fnrnafv
StepHypRef Expression
1 dfafn5a 47110 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
21rneqd 5952 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹'''𝑥)))
3 eqid 2735 . . 3 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
43rnmpt 5971 . 2 ran (𝑥𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}
52, 4eqtrdi 2791 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  {cab 2712  wrex 3068  cmpt 5231  ran crn 5690   Fn wfn 6558  '''cafv 47067
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-11 2155  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-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070
This theorem is referenced by:  afvelrnb  47113  afvelrnb0  47114
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