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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 6582 | . . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) | |
2 | f1f1orn 6626 | . . 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 5568 | . . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶) | |
5 | f1oeq3 6606 | . . 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 236 | 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 208 ∧ wa 398 = wceq 1537 ⊆ wss 3936 ran crn 5556 ↾ cres 5557 “ cima 5558 –1-1→wf1 6352 –1-1-onto→wf1o 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 |
This theorem is referenced by: f1imacnv 6631 f1oresrab 6889 isores3 7088 isoini2 7092 f1imaeng 8569 f1imaen2g 8570 domunsncan 8617 php3 8703 ssfi 8738 infdifsn 9120 infxpenlem 9439 ackbij2lem2 9662 fin1a2lem6 9827 grothomex 10251 fsumss 15082 ackbijnn 15183 fprodss 15302 unbenlem 16244 eqgen 18333 symgfixelsi 18563 gsumval3lem1 19025 gsumval3lem2 19026 gsumzaddlem 19041 coe1mul2lem2 20436 lindsmm 20972 tsmsf1o 22753 ovoliunlem1 24103 dvcnvrelem2 24615 logf1o2 25233 dvlog 25234 ushgredgedg 27011 ushgredgedgloop 27013 trlreslem 27481 adjbd1o 29862 rinvf1o 30375 padct 30455 indf1ofs 31285 eulerpartgbij 31630 eulerpartlemgh 31636 ballotlemfrc 31784 reprpmtf1o 31897 erdsze2lem2 32451 poimirlem4 34911 poimirlem9 34916 ismtyres 35101 pwfi2f1o 39716 sge0f1o 42684 f1oresf1o 43509 |
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