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Theorem dfafn5a 39889
 Description: Representation of a function in terms of its values, analogous to dffn5 6151 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfafn5a (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dfafn5a
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnrel 5903 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 5503 . . . 4 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 207 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 5907 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 449 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 665 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2617 . . . . . . 7 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
8 fnbrafvb 39883 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))
97, 8syl5bb 271 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹'''𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 671 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹'''𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 270 . . . 4 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹'''𝑥))))
1211opabbidv 4648 . . 3 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
133, 12eqtrd 2644 . 2 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
14 df-mpt 4645 . 2 (𝑥𝐴 ↦ (𝐹'''𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))}
1513, 14syl6eqr 2662 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  Rel wrel 5043   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-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:  dfafn5b  39890  fnrnafv  39891
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