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Theorem dfafn5a 43709
Description: Representation of a function in terms of its values, analogous to dffn5 6703 (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 6428 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 6018 . . . 4 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 221 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 6434 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 416 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 566 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2808 . . . . . . 7 (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝑦)
8 fnbrafvb 43703 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹'''𝑥) = 𝑦𝑥𝐹𝑦))
97, 8syl5bb 286 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹'''𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 582 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹'''𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 285 . . . 4 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹'''𝑥))))
1211opabbidv 5099 . . 3 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
133, 12eqtrd 2836 . 2 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))})
14 df-mpt 5114 . 2 (𝑥𝐴 ↦ (𝐹'''𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹'''𝑥))}
1513, 14eqtr4di 2854 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112   class class class wbr 5033  {copab 5095  cmpt 5113  Rel wrel 5528   Fn wfn 6323  '''cafv 43666
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-aiota 43635  df-dfat 43668  df-afv 43669
This theorem is referenced by:  dfafn5b  43710  fnrnafv  43711
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