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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 5788 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 6010 | . 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 5078 ◡ccnv 5587 Rel wrel 5593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-sb 2071 df-clab 2717 df-cleq 2731 df-clel 2817 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-br 5079 df-opab 5141 df-xp 5594 df-rel 5595 df-cnv 5596 |
This theorem is referenced by: compssiso 10114 fneval 34520 brcnvep 36383 brid 36421 brcnvrabga 36456 br1cnvxrn2 36501 br1cnvssrres 36602 brcnvssr 36603 brco2f1o 41595 brco3f1o 41596 neicvgnvor 41679 |
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