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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 5541 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 5749 | . 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 198 class class class wbr 4875 ◡ccnv 5345 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 |
This theorem is referenced by: compssiso 9518 fneval 32880 brcnvep 34578 brid 34621 brcnvrabga 34653 br1cnvxrn2 34697 br1cnvssrres 34798 brcnvssr 34799 brco2f1o 39165 brco3f1o 39166 neicvgnvor 39249 |
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