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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 6811 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | |
| 2 | f1f1orn 6859 | . . 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 5698 | . . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶) | |
| 5 | f1oeq3 6838 | . . 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 1540 ⊆ wss 3951 ran crn 5686 ↾ cres 5687 “ cima 5688 –1-1→wf1 6558 –1-1-onto→wf1o 6560 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 |
| This theorem is referenced by: f1imacnv 6864 f1oresrab 7147 f1ocoima 7323 isores3 7355 isoini2 7359 f1imaeng 9054 f1imaen2g 9055 f1imaen3g 9056 domunsncan 9112 ssfiALT 9214 f1imaenfi 9235 php3 9249 php3OLD 9261 infdifsn 9697 infxpenlem 10053 ackbij2lem2 10279 fin1a2lem6 10445 grothomex 10869 fsumss 15761 ackbijnn 15864 fprodss 15984 unbenlem 16946 eqgen 19199 symgfixelsi 19453 gsumval3lem1 19923 gsumval3lem2 19924 gsumzaddlem 19939 lindsmm 21848 coe1mul2lem2 22271 tsmsf1o 24153 ovoliunlem1 25537 dvcnvrelem2 26057 logf1o2 26692 dvlog 26693 ushgredgedg 29246 ushgredgedgloop 29248 trlreslem 29717 adjbd1o 32104 rinvf1o 32640 padct 32731 indf1ofs 32851 eulerpartgbij 34374 eulerpartlemgh 34380 ballotlemfrc 34529 reprpmtf1o 34641 erdsze2lem2 35209 poimirlem4 37631 poimirlem9 37636 ismtyres 37815 pwfi2f1o 43108 sge0f1o 46397 3f1oss1 47087 f1oresf1o 47302 uhgrimisgrgric 47899 |
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