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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 | ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ssres 6812 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | |
2 | f1f1orn 6860 | . . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶)) |
4 | df-ima 5702 | . . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶) | |
5 | f1oeq3 6839 | . . 3 ⊢ ((𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶) → ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) ↔ (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) ↔ (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶)) |
7 | 3, 6 | sylibr 234 | 1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3963 ran crn 5690 ↾ cres 5691 “ cima 5692 –1-1→wf1 6560 –1-1-onto→wf1o 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: f1imacnv 6865 f1oresrab 7147 f1ocoima 7323 isores3 7355 isoini2 7359 f1imaeng 9053 f1imaen2g 9054 f1imaen3g 9055 domunsncan 9111 ssfiALT 9213 f1imaenfi 9233 php3 9247 php3OLD 9259 infdifsn 9695 infxpenlem 10051 ackbij2lem2 10277 fin1a2lem6 10443 grothomex 10867 fsumss 15758 ackbijnn 15861 fprodss 15981 unbenlem 16942 eqgen 19212 symgfixelsi 19468 gsumval3lem1 19938 gsumval3lem2 19939 gsumzaddlem 19954 lindsmm 21866 coe1mul2lem2 22287 tsmsf1o 24169 ovoliunlem1 25551 dvcnvrelem2 26072 logf1o2 26707 dvlog 26708 ushgredgedg 29261 ushgredgedgloop 29263 trlreslem 29732 adjbd1o 32114 rinvf1o 32647 padct 32737 indf1ofs 34007 eulerpartgbij 34354 eulerpartlemgh 34360 ballotlemfrc 34508 reprpmtf1o 34620 erdsze2lem2 35189 poimirlem4 37611 poimirlem9 37616 ismtyres 37795 pwfi2f1o 43085 sge0f1o 46338 3f1oss1 47025 f1oresf1o 47240 uhgrimisgrgric 47837 |
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