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Theorem dfafn5b 41562
Description: Representation of a function in terms of its values, analogous to dffn5 6280 (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 41561 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
2 eqid 2651 . . . 4 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
32fnmpt 6058 . . 3 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)
4 fneq1 6017 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴))
53, 4syl5ibrcom 237 . 2 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)) → 𝐹 Fn 𝐴))
61, 5impbid2 216 1 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wral 2941  cmpt 4762   Fn wfn 5921  '''cafv 41515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-dfat 41517  df-afv 41518
This theorem is referenced by: (None)
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