|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > brres | Structured version Visualization version GIF version | ||
| Description: Binary relation on a restriction. (Contributed by Mario Carneiro, 4-Nov-2015.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| brres | ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opelres 6002 | . 2 ⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | |
| 2 | df-br 5143 | . 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴)) | |
| 3 | df-br 5143 | . . 3 ⊢ (𝐵𝑅𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝑅) | |
| 4 | 3 | anbi2i 623 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅)) | 
| 5 | 1, 2, 4 | 3bitr4g 314 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 ↾ cres 5686 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-res 5696 | 
| This theorem is referenced by: brresi 6005 dfima2 6079 predres 6359 ttrclselem2 9767 axhcompl-zf 31018 fv1stcnv 35778 fv2ndcnv 35779 bj-idreseq 37164 bj-idreseqb 37165 brcnvepres 38269 brres2 38270 eldmres 38272 elrnres 38273 elecres 38279 brinxprnres 38293 exanres 38297 eqres 38342 alrmomorn 38360 alrmomodm 38361 brxrn 38376 rnxrnres 38401 1cossres 38431 brressn 38443 eldm1cossres 38462 brssrres 38506 disjres 38746 antisymrelres 38765 dfdfat2 47145 | 
| Copyright terms: Public domain | W3C validator |