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Mirrors > Home > MPE Home > Th. List > f1oi | Structured version Visualization version Unicode version |
Description: A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
f1oi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 6008 | . 2 | |
2 | cnvresid 5968 | . . . 4 | |
3 | 2 | fneq1i 5985 | . . 3 |
4 | 1, 3 | mpbir 221 | . 2 |
5 | dff1o4 6145 | . 2 | |
6 | 1, 4, 5 | mpbir2an 955 | 1 |
Colors of variables: wff setvar class |
Syntax hints: cid 5023 ccnv 5113 cres 5116 wfn 5883 wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: f1ovi 6175 fveqf1o 6557 isoid 6579 enrefg 7987 ssdomg 8001 hartogslem1 8447 wdomref 8477 infxpenc 8841 pwfseqlem5 9485 dfle2 11980 fproddvdsd 15059 wunndx 15878 idfucl 16541 idffth 16593 ressffth 16598 setccatid 16734 estrccatid 16772 funcestrcsetclem7 16786 funcestrcsetclem8 16787 equivestrcsetc 16792 funcsetcestrclem7 16801 funcsetcestrclem8 16802 idmhm 17344 idghm 17675 symggrp 17820 symgid 17821 idresperm 17829 islinds2 20152 lindfres 20162 lindsmm 20167 mdetunilem9 20426 ssidcn 21059 resthauslem 21167 sshauslem 21176 hausdiag 21448 idqtop 21509 fmid 21764 iducn 22087 mbfid 23403 dvid 23681 dvexp 23716 wilthlem2 24795 wilthlem3 24796 idmot 25432 ausgrusgrb 26060 upgrres1 26205 umgrres1 26206 usgrres1 26207 usgrexilem 26336 sizusglecusglem1 26357 pliguhgr 27338 hoif 28613 idunop 28837 idcnop 28840 elunop2 28872 fcobijfs 29501 pmtridf1o 29856 qqhre 30064 rrhre 30065 subfacp1lem4 31165 subfacp1lem5 31166 poimirlem15 33424 poimirlem22 33431 idlaut 35382 tendoidcl 36057 tendo0co2 36076 erng1r 36283 dvalveclem 36314 dva0g 36316 dvh0g 36400 mzpresrename 37313 eldioph2lem1 37323 eldioph2lem2 37324 diophren 37377 kelac2 37635 lnrfg 37689 uspgrsprfo 41756 idmgmhm 41788 funcringcsetcALTV2lem8 42043 funcringcsetclem8ALTV 42066 |
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