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Theorem fnrnafv 46168
Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6950. (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 46166 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
21rneqd 5936 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹'''𝑥)))
3 eqid 2730 . . 3 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
43rnmpt 5953 . 2 ran (𝑥𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}
52, 4eqtrdi 2786 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  {cab 2707  wrex 3068  cmpt 5230  ran crn 5676   Fn wfn 6537  '''cafv 46123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-aiota 46091  df-dfat 46125  df-afv 46126
This theorem is referenced by:  afvelrnb  46169  afvelrnb0  46170
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