Theorem dffn5 6718

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 6448	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		dfrel4v 6041	. . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
3	1, 2	sylib 220	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4		fnbr 6453	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴)
5	4	ex 415	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴))
6	5	pm4.71rd 565	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
7		eqcom 2828	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
8		fnbrfvb 6712	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦))
9	7, 8	syl5bb 285	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦))
10	9	pm5.32da 581	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦)))
11	6, 10	bitr4d 284	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))))
12	11	opabbidv 5124	. . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
13	3, 12	eqtrd 2856	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
14		df-mpt 5139	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
15	13, 14	syl6eqr 2874	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
16		fvex 6677	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
17		eqid 2821	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
18	16, 17	fnmpti 6485	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴
19		fneq1 6438	. . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴))
20	18, 19	mpbiri 260	. 2 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → 𝐹 Fn 𝐴)
21	15, 20	impbii 211	1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   class class class wbr 5058  {copab 5120   ↦ cmpt 5138  Rel wrel 5554   Fn wfn 6344  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by:  fnrnfv  6719  feqmptd  6727  dffn5f  6730  eqfnfv  6796  fndmin  6809  fcompt  6889  funiun  6903  resfunexg  6972  eufnfv  6985  nvocnv  7032  fnov  7276  offveqb  7425  caofinvl  7430  oprabco  7785  df1st2  7787  df2nd2  7788  curry1  7793  curry2  7796  resixpfo  8494  pw2f1olem  8615  marypha2lem3  8895  seqof  13421  prmrec  16252  prdsbascl  16750  xpsaddlem  16840  xpsvsca  16844  oppccatid  16983  fuclid  17230  fucrid  17231  curfuncf  17482  yonedainv  17525  yonffthlem  17526  prdsidlem  17937  pws0g  17941  prdsinvlem  18202  gsummptmhm  19054  staffn  19614  prdslmodd  19735  ofco2  21054  1mavmul  21151  cnmpt1st  22270  cnmpt2nd  22271  ptunhmeo  22410  xpsxmetlem  22983  xpsmet  22986  itg2split  24344  pserulm  25004  pserdvlem2  25010  logcn  25224  logblog  25364  emcllem5  25571  gamcvg2lem  25630  crctcshlem4  27592  eucrct2eupth  28018  fcomptf  30397  gsummpt2d  30682  pl1cn  31193  esumpcvgval  31332  esumcvgsum  31342  eulerpartgbij  31625  dstfrvclim1  31730  ptpconn  32475  knoppcnlem8  33834  knoppcnlem11  33837  ctbssinf  34681  curfv  34866  ovoliunnfl  34928  voliunnfl  34930  fnopabco  34992  upixp  34998  prdsbnd  35065  prdstotbnd  35066  prdsbnd2  35067  fgraphopab  39803  expgrowthi  40658  expgrowth  40660  uzmptshftfval  40671  dvcosre  42189  fourierdlem56  42441  fourierdlem62  42447  fundcmpsurbijinjpreimafv  43561  fundcmpsurinjimaid  43565  fdmdifeqresdif  44384  offvalfv  44385