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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 5948 | . 2 ⊢ (𝐶 ∈ 𝑉 → (⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))) | |
2 | df-br 5111 | . 2 ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑅 ↾ 𝐴)) | |
3 | df-br 5111 | . . 3 ⊢ (𝐵𝑅𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝑅) | |
4 | 3 | anbi2i 624 | . 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 205 ∧ wa 397 ∈ wcel 2107 ⟨cop 4597 class class class wbr 5110 ↾ cres 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-res 5650 |
This theorem is referenced by: brresi 5951 dfima2 6020 predres 6298 ttrclselem2 9669 axhcompl-zf 29982 fv1stcnv 34390 fv2ndcnv 34391 bj-idreseq 35662 bj-idreseqb 35663 brcnvepres 36756 brres2 36757 eldmres 36759 elrnres 36760 elecres 36766 brinxprnres 36781 exanres 36785 eqres 36830 alrmomorn 36848 alrmomodm 36849 brxrn 36865 rnxrnres 36890 1cossres 36920 brressn 36932 eldm1cossres 36951 brssrres 36995 disjres 37235 antisymrelres 37254 dfdfat2 45434 |
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