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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 5111 | . 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 5596 | . . 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 3935 class class class wbr 5059 × cxp 5548 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
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 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-xp 5556 |
This theorem is referenced by: brinxp 5625 opelinxp 5626 fncnv 6422 erinxp 8365 fpwwe2lem8 10053 fpwwe2lem9 10054 fpwwe2lem12 10057 nqerf 10346 nqerid 10349 isstruct 16490 pwsle 16759 psss 17818 psssdm2 17819 pi1cpbl 23642 pi1grplem 23647 brres2 35523 inxpss 35563 inxpss3 35565 idinxpssinxp2 35569 inxp2 35613 inxpxrn 35637 |
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