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Mirrors > Home > MPE Home > Th. List > brinxp2 | Structured version Visualization version GIF version |
Description: Intersection with cross product binary relation. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1085. (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 5117 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ (𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷)) | |
2 | ancom 463 | . 2 ⊢ ((𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷) ↔ (𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷)) | |
3 | brxp 5600 | . . 3 ⊢ (𝐶(𝐴 × 𝐵)𝐷 ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
4 | 3 | anbi1i 625 | . 2 ⊢ ((𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) |
5 | 1, 2, 4 | 3bitri 299 | 1 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∩ cin 3934 class class class wbr 5065 × cxp 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-xp 5560 |
This theorem is referenced by: brinxp 5629 opelinxp 5630 fncnv 6426 erinxp 8370 fpwwe2lem8 10058 fpwwe2lem9 10059 fpwwe2lem12 10062 nqerf 10351 nqerid 10354 isstruct 16495 pwsle 16764 psss 17823 psssdm2 17824 pi1cpbl 23647 pi1grplem 23652 brres2 35528 inxpss 35568 inxpss3 35570 idinxpssinxp2 35574 inxp2 35618 inxpxrn 35642 |
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