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Theorem dffn5im 5144
Description: Representation of a function in terms of its values. The converse holds given the law of the excluded middle; as it is we have most of the converse via funmpt 4864 and dmmptss 4744. (Contributed by Jim Kingdon, 31-Dec-2018.)
Assertion
Ref Expression
dffn5im (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
Distinct variable groups:   x,A   x,𝐹

Proof of Theorem dffn5im
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 fnrel 4923 . . . 4 (𝐹 Fn A → Rel 𝐹)
2 dfrel4v 4699 . . . 4 (Rel 𝐹𝐹 = {⟨x, y⟩ ∣ x𝐹y})
31, 2sylib 127 . . 3 (𝐹 Fn A𝐹 = {⟨x, y⟩ ∣ x𝐹y})
4 fnbr 4927 . . . . . . 7 ((𝐹 Fn A x𝐹y) → x A)
54ex 108 . . . . . 6 (𝐹 Fn A → (x𝐹yx A))
65pm4.71rd 374 . . . . 5 (𝐹 Fn A → (x𝐹y ↔ (x A x𝐹y)))
7 eqcom 2024 . . . . . . 7 (y = (𝐹x) ↔ (𝐹x) = y)
8 fnbrfvb 5139 . . . . . . 7 ((𝐹 Fn A x A) → ((𝐹x) = yx𝐹y))
97, 8syl5bb 181 . . . . . 6 ((𝐹 Fn A x A) → (y = (𝐹x) ↔ x𝐹y))
109pm5.32da 428 . . . . 5 (𝐹 Fn A → ((x A y = (𝐹x)) ↔ (x A x𝐹y)))
116, 10bitr4d 180 . . . 4 (𝐹 Fn A → (x𝐹y ↔ (x A y = (𝐹x))))
1211opabbidv 3797 . . 3 (𝐹 Fn A → {⟨x, y⟩ ∣ x𝐹y} = {⟨x, y⟩ ∣ (x A y = (𝐹x))})
133, 12eqtrd 2054 . 2 (𝐹 Fn A𝐹 = {⟨x, y⟩ ∣ (x A y = (𝐹x))})
14 df-mpt 3794 . 2 (x A ↦ (𝐹x)) = {⟨x, y⟩ ∣ (x A y = (𝐹x))}
1513, 14syl6eqr 2072 1 (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374   class class class wbr 3738  {copab 3791  cmpt 3792  Rel wrel 4277   Fn wfn 4824  cfv 4829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  fnrnfv  5145  feqmptd  5151  dffn5imf  5153  eqfnfv  5190  fndmin  5199  fcompt  5258  resfunexg  5307  eufnfv  5314  fnovim  5532  offveqb  5653  caofinvl  5656  oprabco  5761  df1st2  5763  df2nd2  5764
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