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Mirrors > Home > MPE Home > Th. List > dffn5 | Structured version Visualization version GIF version |
Description: Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dffn5 | ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
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 219 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
4 | fnbr 6453 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
5 | 4 | ex 413 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
6 | 5 | pm4.71rd 563 | . . . . . 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 284 | . . . . . . 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 283 | . . . . 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 259 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → 𝐹 Fn 𝐴) |
21 | 15, 20 | impbii 210 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 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 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 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 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 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 6795 fndmin 6808 fcompt 6888 funiun 6902 resfunexg 6970 eufnfv 6983 nvocnv 7029 fnov 7271 offveqb 7420 caofinvl 7425 oprabco 7782 df1st2 7784 df2nd2 7785 curry1 7790 curry2 7793 resixpfo 8489 pw2f1olem 8610 marypha2lem3 8890 seqof 13417 prmrec 16248 prdsbascl 16746 xpsaddlem 16836 xpsvsca 16840 oppccatid 16979 fuclid 17226 fucrid 17227 curfuncf 17478 yonedainv 17521 yonffthlem 17522 prdsidlem 17933 pws0g 17937 prdsinvlem 18148 gsummptmhm 18991 staffn 19551 prdslmodd 19672 ofco2 20990 1mavmul 21087 cnmpt1st 22206 cnmpt2nd 22207 ptunhmeo 22346 xpsxmetlem 22918 xpsmet 22921 itg2split 24279 pserulm 24939 pserdvlem2 24945 logcn 25157 logblog 25297 emcllem5 25505 gamcvg2lem 25564 crctcshlem4 27526 eucrct2eupth 27952 fcomptf 30332 gsummpt2d 30615 pl1cn 31098 esumpcvgval 31237 esumcvgsum 31247 eulerpartgbij 31530 dstfrvclim1 31635 ptpconn 32378 knoppcnlem8 33737 knoppcnlem11 33740 ctbssinf 34570 curfv 34754 ovoliunnfl 34816 voliunnfl 34818 fnopabco 34881 upixp 34887 prdsbnd 34954 prdstotbnd 34955 prdsbnd2 34956 fgraphopab 39690 expgrowthi 40545 expgrowth 40547 uzmptshftfval 40558 dvcosre 42076 fourierdlem56 42328 fourierdlem62 42334 fdmdifeqresdif 44288 offvalfv 44289 |
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