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Theorem dffn5 6383
Description: Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dffn5 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dffn5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnrel 6129 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 5725 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 208 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 6133 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 397 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 552 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2778 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
8 fnbrfvb 6377 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
97, 8syl5bb 272 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 568 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 271 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
1211opabbidv 4850 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
133, 12eqtrd 2805 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
14 df-mpt 4864 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
1513, 14syl6eqr 2823 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
16 fvex 6342 . . . 4 (𝐹𝑥) ∈ V
17 eqid 2771 . . . 4 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
1816, 17fnmpti 6162 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) Fn 𝐴
19 fneq1 6119 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥𝐴 ↦ (𝐹𝑥)) Fn 𝐴))
2018, 19mpbiri 248 . 2 (𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)) → 𝐹 Fn 𝐴)
2115, 20impbii 199 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145   class class class wbr 4786  {copab 4846  cmpt 4863  Rel wrel 5254   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  fnrnfv  6384  feqmptd  6391  dffn5f  6394  eqfnfv  6454  fndmin  6467  fcompt  6543  funiun  6555  resfunexg  6623  eufnfv  6634  nvocnv  6680  fnov  6915  offveqb  7066  caofinvl  7071  oprabco  7412  df1st2  7414  df2nd2  7415  curry1  7420  curry2  7423  resixpfo  8100  pw2f1olem  8220  marypha2lem3  8499  seqof  13065  prmrec  15833  prdsbascl  16351  xpsaddlem  16443  xpsvsca  16447  oppccatid  16586  fuclid  16833  fucrid  16834  curfuncf  17086  yonedainv  17129  yonffthlem  17130  prdsidlem  17530  pws0g  17534  prdsinvlem  17732  gsummptmhm  18547  staffn  19059  prdslmodd  19182  ofco2  20475  1mavmul  20572  cnmpt1st  21692  cnmpt2nd  21693  ptunhmeo  21832  xpsxmetlem  22404  xpsmet  22407  itg2split  23736  pserulm  24396  pserdvlem2  24402  logcn  24614  logblog  24751  emcllem5  24947  gamcvg2lem  25006  crctcshlem4  26948  eucrct2eupth  27425  fcomptf  29798  gsummpt2d  30121  pl1cn  30341  esumpcvgval  30480  esumcvgsum  30490  eulerpartgbij  30774  dstfrvclim1  30879  ptpconn  31553  knoppcnlem8  32827  knoppcnlem11  32830  curfv  33722  ovoliunnfl  33784  voliunnfl  33786  fnopabco  33849  upixp  33856  prdsbnd  33924  prdstotbnd  33925  prdsbnd2  33926  fgraphopab  38314  expgrowthi  39058  expgrowth  39060  uzmptshftfval  39071  dvcosre  40644  fourierdlem56  40896  fourierdlem62  40902  fdmdifeqresdif  42648  offvalfv  42649
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