Theorem dffn5 6900

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 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem dffn5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrel 6602	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		dfrel4v 6156	. . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
3	1, 2	sylib 218	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4		fnbr 6608	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
5	4	ex 412	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
6	5	pm4.71rd 562	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
7		eqcom 2744	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
8		fnbrfvb 6892	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
9	7, 8	bitrid 283	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
10	9	pm5.32da 579	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
11	6, 10	bitr4d 282	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))))
12	11	opabbidv 5166	. . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
13	3, 12	eqtrd 2772	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
14		df-mpt 5182	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
15	13, 14	eqtr4di 2790	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
16		fvex 6855	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
17		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
18	16, 17	fnmpti 6643	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴
19		fneq1 6591	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴))
20	18, 19	mpbiri 258	. 2 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → 𝐹 Fn 𝐴)
21	15, 20	impbii 209	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  Rel wrel 5637   Fn wfn 6495  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  fnrnfv  6901  feqmptd  6910  dffn5f  6913  eqfnfv  6985  fndmin  6999  fcompt  7088  funiun  7102  resfunexg  7171  eufnfv  7185  nvocnv  7237  fnov  7499  offvalfv  7654  offveqb  7659  caofinvl  7664  oprabco  8048  df1st2  8050  df2nd2  8051  curry1  8056  curry2  8059  resixpfo  8886  pw2f1olem  9021  marypha2lem3  9352  seqof  13994  prmrec  16862  prdsbascl  17415  xpsaddlem  17506  xpsvsca  17510  oppccatid  17654  fuclid  17905  fucrid  17906  curfuncf  18173  yonedainv  18216  yonffthlem  18217  prdsidlem  18706  pws0g  18710  prdsinvlem  18991  gsummptmhm  19881  staffn  20788  prdslmodd  20932  ofco2  22407  1mavmul  22504  cnmpt1st  23624  cnmpt2nd  23625  ptunhmeo  23764  xpsxmetlem  24335  xpsmet  24338  itg2split  25718  pserulm  26399  pserdvlem2  26406  logcn  26624  logblog  26770  emcllem5  26978  gamcvg2lem  27037  crctcshlem4  29905  eucrct2eupth  30332  fcomptf  32747  gsummpt2d  33142  esplyfval3  33748  pl1cn  34132  esumpcvgval  34255  esumcvgsum  34265  eulerpartgbij  34549  dstfrvclim1  34655  ptpconn  35446  knoppcnlem8  36719  knoppcnlem11  36722  ctbssinf  37658  curfv  37848  ovoliunnfl  37910  voliunnfl  37912  fnopabco  37971  upixp  37977  prdsbnd  38041  prdstotbnd  38042  prdsbnd2  38043  sticksstones12a  42524  sticksstones12  42525  sticksstones19  42532  fgraphopab  43557  rp-tfslim  43707  expgrowthi  44686  expgrowth  44688  uzmptshftfval  44699  dvcosre  46267  fourierdlem56  46517  fourierdlem62  46523  fundcmpsurbijinjpreimafv  47764  fundcmpsurinjimaid  47768  fdmdifeqresdif  48699  isnatd  49579