Theorem dffn5 6889

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 6591	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		dfrel4v 6145	. . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
3	1, 2	sylib 218	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4		fnbr 6597	. . . . . . . 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 2740	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
8		fnbrfvb 6881	. . . . . . . 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 5161	. . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
13	3, 12	eqtrd 2768	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
14		df-mpt 5177	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
15	13, 14	eqtr4di 2786	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
16		fvex 6844	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
17		eqid 2733	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
18	16, 17	fnmpti 6632	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴
19		fneq1 6580	. . 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 1541   ∈ wcel 2113   class class class wbr 5095  {copab 5157   ↦ cmpt 5176  Rel wrel 5626   Fn wfn 6484  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497

This theorem is referenced by:  fnrnfv  6890  feqmptd  6899  dffn5f  6902  eqfnfv  6973  fndmin  6987  fcompt  7075  funiun  7089  resfunexg  7158  eufnfv  7172  nvocnv  7224  fnov  7486  offvalfv  7641  offveqb  7646  caofinvl  7651  oprabco  8035  df1st2  8037  df2nd2  8038  curry1  8043  curry2  8046  resixpfo  8870  pw2f1olem  9005  marypha2lem3  9332  seqof  13973  prmrec  16841  prdsbascl  17394  xpsaddlem  17485  xpsvsca  17489  oppccatid  17633  fuclid  17884  fucrid  17885  curfuncf  18152  yonedainv  18195  yonffthlem  18196  prdsidlem  18685  pws0g  18689  prdsinvlem  18970  gsummptmhm  19860  staffn  20767  prdslmodd  20911  ofco2  22386  1mavmul  22483  cnmpt1st  23603  cnmpt2nd  23604  ptunhmeo  23743  xpsxmetlem  24314  xpsmet  24317  itg2split  25697  pserulm  26378  pserdvlem2  26385  logcn  26603  logblog  26749  emcllem5  26957  gamcvg2lem  27016  crctcshlem4  29819  eucrct2eupth  30246  fcomptf  32662  gsummpt2d  33060  esplyfval3  33658  pl1cn  34040  esumpcvgval  34163  esumcvgsum  34173  eulerpartgbij  34457  dstfrvclim1  34563  ptpconn  35349  knoppcnlem8  36616  knoppcnlem11  36619  ctbssinf  37523  curfv  37713  ovoliunnfl  37775  voliunnfl  37777  fnopabco  37836  upixp  37842  prdsbnd  37906  prdstotbnd  37907  prdsbnd2  37908  sticksstones12a  42323  sticksstones12  42324  sticksstones19  42331  fgraphopab  43360  rp-tfslim  43510  expgrowthi  44490  expgrowth  44492  uzmptshftfval  44503  dvcosre  46072  fourierdlem56  46322  fourierdlem62  46328  fundcmpsurbijinjpreimafv  47569  fundcmpsurinjimaid  47573  fdmdifeqresdif  48504  isnatd  49384