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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 6620 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
2 | f1ocnv 6621 | . 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 5548 ran crn 5550 –1-1→wf1 6346 –1-1-onto→wf1o 6348 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 |
This theorem is referenced by: f1dmex 7652 fin1a2lem7 9822 cycpmco2f1 30761 cycpmco2rn 30762 cycpmco2lem2 30764 cycpmco2lem3 30765 cycpmco2lem4 30766 cycpmco2lem5 30767 cycpmco2lem6 30768 cycpmco2lem7 30769 cycpmco2 30770 diophrw 39349 |
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