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Theorem dfafn5b 42010
Description: Representation of a function in terms of its values, analogous to dffn5 6467 (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 42009 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)))
2 eqid 2800 . . . 4 (𝑥𝐴 ↦ (𝐹'''𝑥)) = (𝑥𝐴 ↦ (𝐹'''𝑥))
32fnmpt 6232 . . 3 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝑥𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴)
4 fneq1 6191 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥𝐴 ↦ (𝐹'''𝑥)) Fn 𝐴))
53, 4syl5ibrcom 239 . 2 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥)) → 𝐹 Fn 𝐴))
61, 5impbid2 218 1 (∀𝑥𝐴 (𝐹'''𝑥) ∈ 𝑉 → (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹'''𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  wral 3090  cmpt 4923   Fn wfn 6097  '''cafv 41966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-int 4669  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-res 5325  df-iota 6065  df-fun 6104  df-fn 6105  df-fv 6110  df-aiota 41929  df-dfat 41968  df-afv 41969
This theorem is referenced by: (None)
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