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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 5903 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | dfrel4v 5503 | . . . . 5 ⊢ (Rel 𝐹 ↔ 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) | |
3 | 1, 2 | sylib 207 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦}) |
4 | fnbr 5907 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥𝐹𝑦) → 𝑥 ∈ 𝐴) | |
5 | 4 | ex 449 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 → 𝑥 ∈ 𝐴)) |
6 | 5 | pm4.71rd 665 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
7 | eqcom 2617 | . . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑦) | |
8 | fnbrfvb 6146 | . . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑦 ↔ 𝑥𝐹𝑦)) | |
9 | 7, 8 | syl5bb 271 | . . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑦)) |
10 | 9 | pm5.32da 671 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))) |
11 | 6, 10 | bitr4d 270 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)))) |
12 | 11 | opabbidv 4648 | . . . 4 ⊢ (𝐹 Fn 𝐴 → {〈𝑥, 𝑦〉 ∣ 𝑥𝐹𝑦} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
13 | 3, 12 | eqtrd 2644 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))}) |
14 | df-mpt 4645 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥))} | |
15 | 13, 14 | syl6eqr 2662 | . 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
16 | fvex 6113 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
17 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) | |
18 | 16, 17 | fnmpti 5935 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴 |
19 | fneq1 5893 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) Fn 𝐴)) | |
20 | 18, 19 | mpbiri 247 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) → 𝐹 Fn 𝐴) |
21 | 15, 20 | impbii 198 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 Rel wrel 5043 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: fnrnfv 6152 feqmptd 6159 dffn5f 6162 eqfnfv 6219 fndmin 6232 fcompt 6306 funiun 6318 resfunexg 6384 eufnfv 6395 nvocnv 6437 fnov 6666 offveqb 6817 caofinvl 6822 oprabco 7148 df1st2 7150 df2nd2 7151 curry1 7156 curry2 7159 resixpfo 7832 pw2f1olem 7949 marypha2lem3 8226 seqof 12720 prmrec 15464 prdsbascl 15966 xpsaddlem 16058 xpsvsca 16062 oppccatid 16202 fuclid 16449 fucrid 16450 curfuncf 16701 yonedainv 16744 yonffthlem 16745 prdsidlem 17145 pws0g 17149 prdsinvlem 17347 gsummptmhm 18163 staffn 18672 prdslmodd 18790 ofco2 20076 1mavmul 20173 cnmpt1st 21281 cnmpt2nd 21282 ptunhmeo 21421 xpsxmetlem 21994 xpsmet 21997 itg2split 23322 pserulm 23980 pserdvlem2 23986 logcn 24193 logblog 24330 emcllem5 24526 gamcvg2lem 24585 fcomptf 28840 gsummpt2d 29112 pl1cn 29329 esumpcvgval 29467 esumcvgsum 29477 eulerpartgbij 29761 dstfrvclim1 29866 ptpcon 30469 knoppcnlem8 31660 knoppcnlem11 31663 curfv 32559 ovoliunnfl 32621 voliunnfl 32623 fnopabco 32687 upixp 32694 prdsbnd 32762 prdstotbnd 32763 prdsbnd2 32764 fgraphopab 36807 expgrowthi 37554 expgrowth 37556 uzmptshftfval 37567 dvcosre 38799 fourierdlem56 39055 fourierdlem62 39061 crctcshlem4 41023 eucrct2eupth 41413 fdmdifeqresdif 41913 offvalfv 41914 |
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