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| Mirrors > Home > MPE Home > Th. List > brinxp2 | Structured version Visualization version GIF version | ||
| Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1088. (Revised by Peter Mazsa, 18-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| brinxp2 | ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brin 5194 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ (𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷)) | |
| 2 | ancom 460 | . 2 ⊢ ((𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷) ↔ (𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷)) | |
| 3 | brxp 5733 | . . 3 ⊢ (𝐶(𝐴 × 𝐵)𝐷 ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
| 4 | 3 | anbi1i 624 | . 2 ⊢ ((𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | 
| 5 | 1, 2, 4 | 3bitri 297 | 1 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∩ cin 3949 class class class wbr 5142 × cxp 5682 | 
| 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 | 
| This theorem is referenced by: brinxp 5763 opelinxp 5764 fncnv 6638 erinxp 8832 fpwwe2lem7 10678 fpwwe2lem8 10679 fpwwe2lem11 10682 nqerf 10971 nqerid 10974 isstruct 17190 pwsle 17538 psss 18626 psssdm2 18627 pi1cpbl 25078 pi1grplem 25083 br1cnvinxp 38258 brres2 38270 inxpss 38313 inxpss3 38316 idinxpssinxp2 38320 inxp2 38369 inxpxrn 38397 | 
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