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Theorem dffn5im 5162
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 4881 and dmmptss 4760. (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 4940 . . . 4 (𝐹 Fn A → Rel 𝐹)
2 dfrel4v 4715 . . . 4 (Rel 𝐹𝐹 = {⟨x, y⟩ ∣ x𝐹y})
31, 2sylib 127 . . 3 (𝐹 Fn A𝐹 = {⟨x, y⟩ ∣ x𝐹y})
4 fnbr 4944 . . . . . . 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 2039 . . . . . . 7 (y = (𝐹x) ↔ (𝐹x) = y)
8 fnbrfvb 5157 . . . . . . 7 ((𝐹 Fn A x A) → ((𝐹x) = yx𝐹y))
97, 8syl5bb 181 . . . . . 6 ((𝐹 Fn A x A) → (y = (𝐹x) ↔ x𝐹y))
109pm5.32da 425 . . . . 5 (𝐹 Fn A → ((x A y = (𝐹x)) ↔ (x A x𝐹y)))
116, 10bitr4d 180 . . . 4 (𝐹 Fn A → (x𝐹y ↔ (x A y = (𝐹x))))
1211opabbidv 3814 . . 3 (𝐹 Fn A → {⟨x, y⟩ ∣ x𝐹y} = {⟨x, y⟩ ∣ (x A y = (𝐹x))})
133, 12eqtrd 2069 . 2 (𝐹 Fn A𝐹 = {⟨x, y⟩ ∣ (x A y = (𝐹x))})
14 df-mpt 3811 . 2 (x A ↦ (𝐹x)) = {⟨x, y⟩ ∣ (x A y = (𝐹x))}
1513, 14syl6eqr 2087 1 (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390   class class class wbr 3755  {copab 3808  cmpt 3809  Rel wrel 4293   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fnrnfv  5163  feqmptd  5169  dffn5imf  5171  eqfnfv  5208  fndmin  5217  fcompt  5276  resfunexg  5325  eufnfv  5332  fnovim  5551  offveqb  5672  caofinvl  5675  oprabco  5780  df1st2  5782  df2nd2  5783
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