![]() |
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 5979 | . 2 ⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | |
2 | df-br 5142 | . 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴)) | |
3 | df-br 5142 | . . 3 ⊢ (𝐵𝑅𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝑅) | |
4 | 3 | anbi2i 623 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅)) |
5 | 1, 2, 4 | 3bitr4g 313 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 〈cop 4628 class class class wbr 5141 ↾ cres 5671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-res 5681 |
This theorem is referenced by: brresi 5982 dfima2 6051 predres 6329 ttrclselem2 9703 axhcompl-zf 30114 fv1stcnv 34576 fv2ndcnv 34577 bj-idreseq 35845 bj-idreseqb 35846 brcnvepres 36938 brres2 36939 eldmres 36941 elrnres 36942 elecres 36948 brinxprnres 36963 exanres 36967 eqres 37012 alrmomorn 37030 alrmomodm 37031 brxrn 37047 rnxrnres 37072 1cossres 37102 brressn 37114 eldm1cossres 37133 brssrres 37177 disjres 37417 antisymrelres 37436 dfdfat2 45606 |
Copyright terms: Public domain | W3C validator |