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Mirrors > Home > MPE Home > Th. List > f1ores | Structured version Visualization version GIF version |
Description: The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.) |
Ref | Expression |
---|---|
f1ores | β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1-ontoβ(πΉ β πΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ssres 6792 | . . 3 β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1βπ΅) | |
2 | f1f1orn 6841 | . . 3 β’ ((πΉ βΎ πΆ):πΆβ1-1βπ΅ β (πΉ βΎ πΆ):πΆβ1-1-ontoβran (πΉ βΎ πΆ)) | |
3 | 1, 2 | syl 17 | . 2 β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1-ontoβran (πΉ βΎ πΆ)) |
4 | df-ima 5688 | . . 3 β’ (πΉ β πΆ) = ran (πΉ βΎ πΆ) | |
5 | f1oeq3 6820 | . . 3 β’ ((πΉ β πΆ) = ran (πΉ βΎ πΆ) β ((πΉ βΎ πΆ):πΆβ1-1-ontoβ(πΉ β πΆ) β (πΉ βΎ πΆ):πΆβ1-1-ontoβran (πΉ βΎ πΆ))) | |
6 | 4, 5 | ax-mp 5 | . 2 β’ ((πΉ βΎ πΆ):πΆβ1-1-ontoβ(πΉ β πΆ) β (πΉ βΎ πΆ):πΆβ1-1-ontoβran (πΉ βΎ πΆ)) |
7 | 3, 6 | sylibr 233 | 1 β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1-ontoβ(πΉ β πΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wss 3947 ran crn 5676 βΎ cres 5677 β cima 5678 β1-1βwf1 6537 β1-1-ontoβwf1o 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 |
This theorem is referenced by: f1imacnv 6846 f1oresrab 7121 isores3 7328 isoini2 7332 f1imaeng 9006 f1imaen2g 9007 domunsncan 9068 ssfiALT 9170 f1imaenfi 9194 php3 9208 php3OLD 9220 infdifsn 9648 infxpenlem 10004 ackbij2lem2 10231 fin1a2lem6 10396 grothomex 10820 fsumss 15667 ackbijnn 15770 fprodss 15888 unbenlem 16837 eqgen 19055 symgfixelsi 19297 gsumval3lem1 19767 gsumval3lem2 19768 gsumzaddlem 19783 lindsmm 21374 coe1mul2lem2 21781 tsmsf1o 23640 ovoliunlem1 25010 dvcnvrelem2 25526 logf1o2 26149 dvlog 26150 ushgredgedg 28475 ushgredgedgloop 28477 trlreslem 28945 adjbd1o 31325 rinvf1o 31841 padct 31931 indf1ofs 33012 eulerpartgbij 33359 eulerpartlemgh 33365 ballotlemfrc 33513 reprpmtf1o 33626 erdsze2lem2 34183 poimirlem4 36480 poimirlem9 36485 ismtyres 36664 pwfi2f1o 41823 sge0f1o 45084 f1oresf1o 45984 |
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