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Mirrors > Home > MPE Home > Th. List > relbrcnv | Structured version Visualization version GIF version |
Description: When 𝑅 is a relation, the sethood assumptions on brcnv 5885 can be omitted. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
relbrcnv.1 | ⊢ Rel 𝑅 |
Ref | Expression |
---|---|
relbrcnv | ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relbrcnv.1 | . 2 ⊢ Rel 𝑅 | |
2 | relbrcnvg 6110 | . 2 ⊢ (Rel 𝑅 → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 class class class wbr 5149 ◡ccnv 5677 Rel wrel 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 |
This theorem is referenced by: compssiso 10399 fneval 35964 br1cnvinxp 37855 brcnvep 37864 brid 37905 brcnvrabga 37941 br1cnvxrn2 37995 br1cnvssrres 38104 brcnvssr 38105 brco2f1o 43601 brco3f1o 43602 neicvgnvor 43685 |
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