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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 6458 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | dfrel4v 6033 | . . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) | |
3 | 1, 2 | sylib 221 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
4 | fnbr 6464 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
5 | 4 | ex 416 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
6 | 5 | pm4.71rd 566 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
7 | eqcom 2743 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
8 | fnbrfvb 6743 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
9 | 7, 8 | syl5bb 286 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
10 | 9 | pm5.32da 582 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
11 | 6, 10 | bitr4d 285 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)))) |
12 | 11 | opabbidv 5105 | . . . 4 ⊢ (𝐹 Fn 𝐴 → {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
13 | 3, 12 | eqtrd 2771 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
14 | df-mpt 5121 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} | |
15 | 13, 14 | eqtr4di 2789 | . 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
16 | fvex 6708 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
17 | eqid 2736 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) | |
18 | 16, 17 | fnmpti 6499 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴 |
19 | fneq1 6448 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴)) | |
20 | 18, 19 | mpbiri 261 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → 𝐹 Fn 𝐴) |
21 | 15, 20 | impbii 212 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 {copab 5101 ↦ cmpt 5120 Rel wrel 5541 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: fnrnfv 6750 feqmptd 6758 dffn5f 6761 eqfnfv 6830 fndmin 6843 fcompt 6926 funiun 6940 resfunexg 7009 eufnfv 7023 nvocnv 7070 fnov 7319 offveqb 7471 caofinvl 7476 oprabco 7842 df1st2 7844 df2nd2 7845 curry1 7850 curry2 7853 resixpfo 8595 pw2f1olem 8727 marypha2lem3 9031 seqof 13598 prmrec 16438 prdsbascl 16942 xpsaddlem 17032 xpsvsca 17036 oppccatid 17177 fuclid 17429 fucrid 17430 curfuncf 17700 yonedainv 17743 yonffthlem 17744 prdsidlem 18159 pws0g 18163 prdsinvlem 18426 gsummptmhm 19279 staffn 19839 prdslmodd 19960 ofco2 21302 1mavmul 21399 cnmpt1st 22519 cnmpt2nd 22520 ptunhmeo 22659 xpsxmetlem 23231 xpsmet 23234 itg2split 24601 pserulm 25268 pserdvlem2 25274 logcn 25489 logblog 25629 emcllem5 25836 gamcvg2lem 25895 crctcshlem4 27858 eucrct2eupth 28282 fcomptf 30669 gsummpt2d 30982 pl1cn 31573 esumpcvgval 31712 esumcvgsum 31722 eulerpartgbij 32005 dstfrvclim1 32110 ptpconn 32862 knoppcnlem8 34366 knoppcnlem11 34369 ctbssinf 35263 curfv 35443 ovoliunnfl 35505 voliunnfl 35507 fnopabco 35567 upixp 35573 prdsbnd 35637 prdstotbnd 35638 prdsbnd2 35639 sticksstones12a 39782 sticksstones12 39783 fgraphopab 40679 expgrowthi 41565 expgrowth 41567 uzmptshftfval 41578 dvcosre 43071 fourierdlem56 43321 fourierdlem62 43327 fundcmpsurbijinjpreimafv 44475 fundcmpsurinjimaid 44479 fdmdifeqresdif 45293 offvalfv 45294 |
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