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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 5763 | . 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 206 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 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-in 3958 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: poinxp 5766 soinxp 5767 frinxp 5768 seinxp 5769 exfo 7125 isores2 7353 ltpiord 10927 ordpinq 10983 pwsleval 17538 tsrss 18634 ordtrest 23210 ordtrest2lem 23211 ordtrestNEW 33920 ordtrest2NEWlem 33921 satefvfmla0 35423 | 
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