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Theorem dfafn5a 45287
Description: Representation of a function in terms of its values, analogous to dffn5 6898 (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 6601 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 6140 . . . 4 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 217 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 6607 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 413 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 563 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2743 . . . . . . 7 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
8 fnbrafvb 45281 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))
97, 8bitrid 282 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹'''𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 579 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹'''𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 281 . . . 4 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹'''𝑥))))
1211opabbidv 5169 . . 3 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
133, 12eqtrd 2776 . 2 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
14 df-mpt 5187 . 2 (𝑥𝐴 ↦ (𝐹'''𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))}
1513, 14eqtr4di 2794 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106   class class class wbr 5103  {copab 5165  cmpt 5186  Rel wrel 5636   Fn wfn 6488  '''cafv 45244
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501  df-aiota 45212  df-dfat 45246  df-afv 45247
This theorem is referenced by:  dfafn5b  45288  fnrnafv  45289
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