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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 6747 | . . 3 β’ ((πΉ:π΄β1-1βπ΅ β§ πΆ β π΄) β (πΉ βΎ πΆ):πΆβ1-1βπ΅) | |
2 | f1f1orn 6796 | . . 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 5647 | . . 3 β’ (πΉ β πΆ) = ran (πΉ βΎ πΆ) | |
5 | f1oeq3 6775 | . . 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 3911 ran crn 5635 βΎ cres 5636 β cima 5637 β1-1βwf1 6494 β1-1-ontoβwf1o 6496 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 |
This theorem is referenced by: f1imacnv 6801 f1oresrab 7074 isores3 7281 isoini2 7285 f1imaeng 8955 f1imaen2g 8956 domunsncan 9017 ssfiALT 9119 f1imaenfi 9143 php3 9157 php3OLD 9169 infdifsn 9594 infxpenlem 9950 ackbij2lem2 10177 fin1a2lem6 10342 grothomex 10766 fsumss 15611 ackbijnn 15714 fprodss 15832 unbenlem 16781 eqgen 18984 symgfixelsi 19218 gsumval3lem1 19683 gsumval3lem2 19684 gsumzaddlem 19699 lindsmm 21237 coe1mul2lem2 21642 tsmsf1o 23499 ovoliunlem1 24869 dvcnvrelem2 25385 logf1o2 26008 dvlog 26009 ushgredgedg 28180 ushgredgedgloop 28182 trlreslem 28650 adjbd1o 31030 rinvf1o 31547 padct 31639 indf1ofs 32628 eulerpartgbij 32975 eulerpartlemgh 32981 ballotlemfrc 33129 reprpmtf1o 33242 erdsze2lem2 33801 poimirlem4 36085 poimirlem9 36090 ismtyres 36270 pwfi2f1o 41426 sge0f1o 44630 f1oresf1o 45529 |
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