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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 5854 | . 2 ⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) | |
2 | df-br 5060 | . 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴)) | |
3 | df-br 5060 | . . 3 ⊢ (𝐵𝑅𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝑅) | |
4 | 3 | anbi2i 624 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅)) |
5 | 1, 2, 4 | 3bitr4g 316 | 1 ⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 〈cop 4567 class class class wbr 5059 ↾ cres 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 df-res 5562 |
This theorem is referenced by: brresi 5857 dfima2 5926 axhcompl-zf 28769 fv1stcnv 33015 fv2ndcnv 33016 bj-idreseq 34448 bj-idreseqb 34449 brcnvepres 35522 brres2 35523 eldmres 35524 elecres 35528 brinxprnres 35542 exanres 35546 eqres 35591 alrmomorn 35606 alrmomodm 35607 brxrn 35620 rnxrnres 35641 1cossres 35668 eldm1cossres 35694 brssrres 35738 dfdfat2 43320 |
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