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Mirrors > Home > MPE Home > Th. List > f1cnv | Structured version Visualization version GIF version |
Description: The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.) |
Ref | Expression |
---|---|
f1cnv | ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f1orn 6148 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
2 | f1ocnv 6149 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ◡ccnv 5113 ran crn 5115 –1-1→wf1 5885 –1-1-onto→wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: f1dmex 7136 fin1a2lem7 9228 diophrw 37322 |
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