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Mirrors > Home > MPE Home > Th. List > brinxp | Structured version Visualization version GIF version |
Description: Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.) |
Ref | Expression |
---|---|
brinxp | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brinxp2 5753 | . 2 ⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝐴𝑅𝐵)) | |
2 | 1 | baibr 536 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 ∩ cin 3947 class class class wbr 5148 × cxp 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 |
This theorem is referenced by: poinxp 5756 soinxp 5757 frinxp 5758 seinxp 5759 exfo 7106 isores2 7333 ltpiord 10888 ordpinq 10944 pwsleval 17446 tsrss 18552 ordtrest 23026 ordtrest2lem 23027 ordtrestNEW 33365 ordtrest2NEWlem 33366 satefvfmla0 34873 |
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