Theorem dffn5 6875

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 6578	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		dfrel4v 6132	. . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
3	1, 2	sylib 218	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦})
4		fnbr 6584	. . . . . . . 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 2738	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦)
8		fnbrfvb 6867	. . . . . . . 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 5152	. . . 4 ⊢ (𝐹 Fn 𝐴 → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐹𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
13	3, 12	eqtrd 2766	. . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))})
14		df-mpt 5168	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}
15	13, 14	eqtr4di 2784	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
16		fvex 6830	. . . 4 ⊢ (𝐹‘𝑥) ∈ V
17		eqid 2731	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
18	16, 17	fnmpti 6619	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴
19		fneq1 6567	. . 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 2111   class class class wbr 5086  {copab 5148   ↦ cmpt 5167  Rel wrel 5616   Fn wfn 6471  ‘cfv 6476

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fn 6479  df-fv 6484

This theorem is referenced by:  fnrnfv  6876  feqmptd  6885  dffn5f  6888  eqfnfv  6959  fndmin  6973  fcompt  7061  funiun  7075  resfunexg  7144  eufnfv  7158  nvocnv  7210  fnov  7472  offvalfv  7627  offveqb  7632  caofinvl  7637  oprabco  8021  df1st2  8023  df2nd2  8024  curry1  8029  curry2  8032  resixpfo  8855  pw2f1olem  8989  marypha2lem3  9316  seqof  13961  prmrec  16829  prdsbascl  17382  xpsaddlem  17472  xpsvsca  17476  oppccatid  17620  fuclid  17871  fucrid  17872  curfuncf  18139  yonedainv  18182  yonffthlem  18183  prdsidlem  18672  pws0g  18676  prdsinvlem  18957  gsummptmhm  19847  staffn  20753  prdslmodd  20897  ofco2  22361  1mavmul  22458  cnmpt1st  23578  cnmpt2nd  23579  ptunhmeo  23718  xpsxmetlem  24289  xpsmet  24292  itg2split  25672  pserulm  26353  pserdvlem2  26360  logcn  26578  logblog  26724  emcllem5  26932  gamcvg2lem  26991  crctcshlem4  29793  eucrct2eupth  30217  fcomptf  32632  gsummpt2d  33021  pl1cn  33960  esumpcvgval  34083  esumcvgsum  34093  eulerpartgbij  34377  dstfrvclim1  34483  ptpconn  35269  knoppcnlem8  36534  knoppcnlem11  36537  ctbssinf  37440  curfv  37640  ovoliunnfl  37702  voliunnfl  37704  fnopabco  37763  upixp  37769  prdsbnd  37833  prdstotbnd  37834  prdsbnd2  37835  sticksstones12a  42190  sticksstones12  42191  sticksstones19  42198  fgraphopab  43236  rp-tfslim  43386  expgrowthi  44366  expgrowth  44368  uzmptshftfval  44379  dvcosre  45950  fourierdlem56  46200  fourierdlem62  46206  fundcmpsurbijinjpreimafv  47438  fundcmpsurinjimaid  47442  fdmdifeqresdif  48373  isnatd  49255