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Theorem dfafn5a 47109
Description: Representation of a function in terms of its values, analogous to dffn5 6966 (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 6670 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 6211 . . . 4 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 218 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 6676 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 412 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 562 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2741 . . . . . . 7 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
8 fnbrafvb 47103 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))
97, 8bitrid 283 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹'''𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 579 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹'''𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 282 . . . 4 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹'''𝑥))))
1211opabbidv 5213 . . 3 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
133, 12eqtrd 2774 . 2 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
14 df-mpt 5231 . 2 (𝑥𝐴 ↦ (𝐹'''𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))}
1513, 14eqtr4di 2792 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105   class class class wbr 5147  {copab 5209  cmpt 5230  Rel wrel 5693   Fn wfn 6557  '''cafv 47066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-aiota 47034  df-dfat 47068  df-afv 47069
This theorem is referenced by:  dfafn5b  47110  fnrnafv  47111
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