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Theorem dffn5 6135
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 5888 . . . . 5 (𝐹 Fn 𝐴 → Rel 𝐹)
2 dfrel4v 5488 . . . . 5 (Rel 𝐹𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
31, 2sylib 206 . . . 4 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4 fnbr 5892 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
54ex 448 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
65pm4.71rd 664 . . . . . 6 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑥𝐹𝑦)))
7 eqcom 2616 . . . . . . . 8 (𝑦 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑦)
8 fnbrfvb 6130 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = 𝑦𝑥𝐹𝑦))
97, 8syl5bb 270 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑦 = (𝐹𝑥) ↔ 𝑥𝐹𝑦))
109pm5.32da 670 . . . . . 6 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑦 = (𝐹𝑥)) ↔ (𝑥𝐴𝑥𝐹𝑦)))
116, 10bitr4d 269 . . . . 5 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = (𝐹𝑥))))
1211opabbidv 4642 . . . 4 (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
133, 12eqtrd 2643 . . 3 (𝐹 Fn 𝐴𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))})
14 df-mpt 4639 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = (𝐹𝑥))}
1513, 14syl6eqr 2661 . 2 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
16 fvex 6097 . . . 4 (𝐹𝑥) ∈ V
17 eqid 2609 . . . 4 (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐹𝑥))
1816, 17fnmpti 5920 . . 3 (𝑥𝐴 ↦ (𝐹𝑥)) Fn 𝐴
19 fneq1 5878 . . 3 (𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥𝐴 ↦ (𝐹𝑥)) Fn 𝐴))
2018, 19mpbiri 246 . 2 (𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)) → 𝐹 Fn 𝐴)
2115, 20impbii 197 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976   class class class wbr 4577  {copab 4636  cmpt 4637  Rel wrel 5032   Fn wfn 5784  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fn 5792  df-fv 5797
This theorem is referenced by:  fnrnfv  6136  feqmptd  6143  dffn5f  6146  eqfnfv  6203  fndmin  6216  fcompt  6290  resfunexg  6361  eufnfv  6372  nvocnv  6414  fnov  6643  offveqb  6794  caofinvl  6799  oprabco  7125  df1st2  7127  df2nd2  7128  curry1  7133  curry2  7136  resixpfo  7809  pw2f1olem  7926  marypha2lem3  8203  seqof  12677  prmrec  15412  prdsbascl  15914  xpsaddlem  16006  xpsvsca  16010  oppccatid  16150  fuclid  16397  fucrid  16398  curfuncf  16649  yonedainv  16692  yonffthlem  16693  prdsidlem  17093  pws0g  17097  prdsinvlem  17295  gsummptmhm  18111  staffn  18620  prdslmodd  18738  ofco2  20023  1mavmul  20120  cnmpt1st  21228  cnmpt2nd  21229  ptunhmeo  21368  xpsxmetlem  21941  xpsmet  21944  itg2split  23266  pserulm  23924  pserdvlem2  23930  logcn  24137  logblog  24274  emcllem5  24470  gamcvg2lem  24529  fcomptf  28633  gsummpt2d  28905  pl1cn  29122  esumpcvgval  29260  esumcvgsum  29270  eulerpartgbij  29554  dstfrvclim1  29659  ptpcon  30262  knoppcnlem8  31453  knoppcnlem11  31456  curfv  32342  ovoliunnfl  32404  voliunnfl  32406  fnopabco  32470  upixp  32477  prdsbnd  32545  prdstotbnd  32546  prdsbnd2  32547  fgraphopab  36590  expgrowthi  37337  expgrowth  37339  uzmptshftfval  37350  dvcosre  38582  fourierdlem56  38838  fourierdlem62  38844  funiun  40122  crctcshlem4  41004  eucrct2eupth  41394  fdmdifeqresdif  41894  offvalfv  41895
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