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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 6860 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
2 | f1ocnv 6861 | . 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 5688 ran crn 5690 –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-rab 3434 df-v 3480 df-dif 3966 df-un 3968 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-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 |
This theorem is referenced by: f1dmex 7980 f1domfi 9219 fin1a2lem7 10444 cycpmco2f1 33127 cycpmco2rn 33128 cycpmco2lem2 33130 cycpmco2lem3 33131 cycpmco2lem4 33132 cycpmco2lem5 33133 cycpmco2lem6 33134 cycpmco2lem7 33135 cycpmco2 33136 diophrw 42747 |
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