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Theorem dfafn5b 47178
Description: Representation of a function in terms of its values, analogous to dffn5 6966 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfafn5b (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem dfafn5b
StepHypRef Expression
1 dfafn5a 47177 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
2 eqid 2736 . . . 4 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
32fnmpt 6707 . . 3 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)
4 fneq1 6658 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴))
53, 4syl5ibrcom 247 . 2 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)) → 𝐹 Fn 𝐴))
61, 5impbid2 226 1 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  wral 3060  cmpt 5224   Fn wfn 6555  '''cafv 47134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-aiota 47102  df-dfat 47136  df-afv 47137
This theorem is referenced by: (None)
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