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Mirrors > Home > MPE Home > Th. List > fcoi1 | Structured version Visualization version Unicode version |
Description: Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fcoi1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 6045 | . 2 | |
2 | df-fn 5891 | . . 3 | |
3 | eqimss 3657 | . . . . 5 | |
4 | cnvi 5537 | . . . . . . . . . 10 | |
5 | 4 | reseq1i 5392 | . . . . . . . . 9 |
6 | 5 | cnveqi 5297 | . . . . . . . 8 |
7 | cnvresid 5968 | . . . . . . . 8 | |
8 | 6, 7 | eqtr2i 2645 | . . . . . . 7 |
9 | 8 | coeq2i 5282 | . . . . . 6 |
10 | cores2 5648 | . . . . . 6 | |
11 | 9, 10 | syl5eq 2668 | . . . . 5 |
12 | 3, 11 | syl 17 | . . . 4 |
13 | funrel 5905 | . . . . 5 | |
14 | coi1 5651 | . . . . 5 | |
15 | 13, 14 | syl 17 | . . . 4 |
16 | 12, 15 | sylan9eqr 2678 | . . 3 |
17 | 2, 16 | sylbi 207 | . 2 |
18 | 1, 17 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wss 3574 cid 5023 ccnv 5113 cdm 5114 cres 5116 ccom 5118 wrel 5119 wfun 5882 wfn 5883 wf 5884 |
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 |
This theorem is referenced by: fcof1oinvd 6548 mapen 8124 mapfien 8313 hashfacen 13238 cofurid 16551 setccatid 16734 estrccatid 16772 curf2ndf 16887 symgid 17821 f1omvdco2 17868 psgnunilem1 17913 pf1mpf 19716 pf1ind 19719 wilthlem3 24796 hoico1 28615 fmptco1f1o 29434 fcobijfs 29501 reprpmtf1o 30704 ltrncoidN 35414 trlcoabs2N 36010 trlcoat 36011 cdlemg47a 36022 cdlemg46 36023 trljco 36028 tendo1mulr 36059 tendo0co2 36076 cdlemi2 36107 cdlemk2 36120 cdlemk4 36122 cdlemk8 36126 cdlemk53 36245 cdlemk55a 36247 dvhopN 36405 dihopelvalcpre 36537 dihmeetlem1N 36579 dihglblem5apreN 36580 diophrw 37322 mendring 37762 rngccatidALTV 41989 ringccatidALTV 42052 |
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