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Theorem fnrnafv 45804
Description: The range of a function expressed as a collection of the function's values, analogous to fnrnfv 6947. (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 45802 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
21rneqd 5934 . 2 (𝐹 Fn 𝐴 → ran 𝐹 = ran (𝑥𝐴 ↦ (𝐹'''𝑥)))
3 eqid 2733 . . 3 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
43rnmpt 5951 . 2 ran (𝑥𝐴 ↦ (𝐹'''𝑥)) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}
52, 4eqtrdi 2789 1 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  {cab 2710  wrex 3071  cmpt 5229  ran crn 5675   Fn wfn 6534  '''cafv 45759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-iota 6491  df-fun 6541  df-fn 6542  df-fv 6547  df-aiota 45727  df-dfat 45761  df-afv 45762
This theorem is referenced by:  afvelrnb  45805  afvelrnb0  45806
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