|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6859 | . 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹) | |
| 2 | f1ocnv 6860 | . 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 5684 ran crn 5686 –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-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 | 
| This theorem is referenced by: f1dmex 7981 f1domfi 9221 fin1a2lem7 10446 cycpmco2f1 33144 cycpmco2rn 33145 cycpmco2lem2 33147 cycpmco2lem3 33148 cycpmco2lem4 33149 cycpmco2lem5 33150 cycpmco2lem6 33151 cycpmco2lem7 33152 cycpmco2 33153 diophrw 42770 | 
| Copyright terms: Public domain | W3C validator |