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| Mirrors > Home > MPE Home > Th. List > brxp | Structured version Visualization version GIF version | ||
| Description: Binary relation on a Cartesian product. (Contributed by NM, 22-Apr-2004.) |
| Ref | Expression |
|---|---|
| brxp | ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 5144 | . 2 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
| 2 | opelxp 5721 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
| 3 | 1, 2 | bitri 275 | 1 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: brrelex12 5737 brel 5750 brinxp2 5763 eqbrrdva 5880 ssrelrn 5905 dmxp 5939 xpidtr 6142 xpco 6309 dfpo2 6316 predtrss 6343 isocnv3 7352 tpostpos 8271 brinxper 8774 swoer 8776 erinxp 8831 ecopover 8861 infxpenlem 10053 fpwwe2lem5 10675 fpwwe2lem6 10676 fpwwe2lem8 10678 fpwwe2lem11 10681 fpwwe2lem12 10682 fpwwe2 10683 ltxrlt 11331 ltxr 13157 xpcogend 15013 invfuc 18022 elhoma 18077 efglem 19734 gsumcom3fi 19997 gsumdixp 20316 znleval 21573 gsumbagdiag 21951 psrass1lem 21952 opsrtoslem2 22080 slenlt 27797 brelg 32621 posrasymb 32955 trleile 32961 ecxpid 33389 qusxpid 33391 metider 33893 satefvfmla1 35430 mclsppslem 35588 xpab 35726 dfon3 35893 brbigcup 35899 brsingle 35918 brimage 35927 brcart 35933 brapply 35939 brcup 35940 brcap 35941 funpartlem 35943 dfrdg4 35952 brub 35955 bj-xpcossxp 37190 itg2gt0cn 37682 grucollcld 44279 grumnud 44305 coxp 48744 |
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