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Theorem fnrnafv 43761
 Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6701. (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 43759 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
21rneqd 5773 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹'''𝑥)))
3 eqid 2798 . . 3 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
43rnmpt 5792 . 2 ran (𝑥𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}
52, 4eqtrdi 2849 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  {cab 2776  ∃wrex 3107   ↦ cmpt 5111  ran crn 5521   Fn wfn 6320  '''cafv 43716 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4840  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-iota 6284  df-fun 6327  df-fn 6328  df-fv 6333  df-aiota 43685  df-dfat 43718  df-afv 43719 This theorem is referenced by:  afvelrnb  43762  afvelrnb0  43763
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