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Theorem fnrnafv 41767
 Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6406. (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 41765 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
21rneqd 5509 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹'''𝑥)))
3 eqid 2761 . . 3 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
43rnmpt 5527 . 2 ran (𝑥𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}
52, 4syl6eq 2811 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632  {cab 2747  ∃wrex 3052   ↦ cmpt 4882  ran crn 5268   Fn wfn 6045  '''cafv 41719 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-iota 6013  df-fun 6052  df-fn 6053  df-fv 6058  df-dfat 41721  df-afv 41722 This theorem is referenced by:  afvelrnb  41768  afvelrnb0  41769
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