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Theorem afvelrnb 39892
Description: A member of a function's range is a value of the function, analogous to fvelrnb 6153 with the additional requirement that the member must be a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvelrnb ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem afvelrnb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnrnafv 39891 . . . 4 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
21adantr 480 . . 3 ((𝐹 Fn 𝐴𝐵𝑉) → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
32eleq2d 2673 . 2 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}))
4 eqeq1 2614 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
5 eqcom 2617 . . . . . 6 (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
64, 5syl6bb 275 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
76rexbidv 3034 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
87elabg 3320 . . 3 (𝐵𝑉 → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
98adantl 481 . 2 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
103, 9bitrd 267 1 ((𝐹 Fn 𝐴𝐵𝑉) → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wrex 2897  ran crn 5039   Fn wfn 5799  '''cafv 39843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-dfat 39845  df-afv 39846
This theorem is referenced by: (None)
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