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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 5412 | . 2 ⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝐴𝑅𝐵)) | |
2 | 1 | baibr 534 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 ∩ cin 3796 class class class wbr 4872 × cxp 5339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pr 5126 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-br 4873 df-opab 4935 df-xp 5347 |
This theorem is referenced by: poinxp 5416 soinxp 5417 frinxp 5418 seinxp 5419 exfo 6625 isores2 6837 ltpiord 10023 ordpinq 10079 pwsleval 16505 tsrss 17575 ordtrest 21376 ordtrest2lem 21377 ordtrestNEW 30511 ordtrest2NEWlem 30512 |
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