![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6796 | . . 3 β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1βπ΅) | |
2 | f1f1orn 6845 | . . 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 5690 | . . 3 β’ (πΉ β πΆ) = ran (πΉ βΎ πΆ) | |
5 | f1oeq3 6824 | . . 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 397 = wceq 1542 β wss 3949 ran crn 5678 βΎ cres 5679 β cima 5680 β1-1βwf1 6541 β1-1-ontoβwf1o 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 |
This theorem is referenced by: f1imacnv 6850 f1oresrab 7125 isores3 7332 isoini2 7336 f1imaeng 9010 f1imaen2g 9011 domunsncan 9072 ssfiALT 9174 f1imaenfi 9198 php3 9212 php3OLD 9224 infdifsn 9652 infxpenlem 10008 ackbij2lem2 10235 fin1a2lem6 10400 grothomex 10824 fsumss 15671 ackbijnn 15774 fprodss 15892 unbenlem 16841 eqgen 19061 symgfixelsi 19303 gsumval3lem1 19773 gsumval3lem2 19774 gsumzaddlem 19789 lindsmm 21383 coe1mul2lem2 21790 tsmsf1o 23649 ovoliunlem1 25019 dvcnvrelem2 25535 logf1o2 26158 dvlog 26159 ushgredgedg 28486 ushgredgedgloop 28488 trlreslem 28956 adjbd1o 31338 rinvf1o 31854 padct 31944 indf1ofs 33024 eulerpartgbij 33371 eulerpartlemgh 33377 ballotlemfrc 33525 reprpmtf1o 33638 erdsze2lem2 34195 poimirlem4 36492 poimirlem9 36497 ismtyres 36676 pwfi2f1o 41838 sge0f1o 45098 f1oresf1o 45998 |
Copyright terms: Public domain | W3C validator |