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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 1086. (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 5082 | . 2 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ (𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷)) | |
2 | ancom 464 | . 2 ⊢ ((𝐶𝑅𝐷 ∧ 𝐶(𝐴 × 𝐵)𝐷) ↔ (𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷)) | |
3 | brxp 5565 | . . 3 ⊢ (𝐶(𝐴 × 𝐵)𝐷 ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) | |
4 | 3 | anbi1i 626 | . 2 ⊢ ((𝐶(𝐴 × 𝐵)𝐷 ∧ 𝐶𝑅𝐷) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) |
5 | 1, 2, 4 | 3bitri 300 | 1 ⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∩ cin 3880 class class class wbr 5030 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 |
This theorem is referenced by: brinxp 5594 opelinxp 5595 fncnv 6397 erinxp 8354 fpwwe2lem8 10048 fpwwe2lem9 10049 fpwwe2lem12 10052 nqerf 10341 nqerid 10344 isstruct 16488 pwsle 16757 psss 17816 psssdm2 17817 pi1cpbl 23649 pi1grplem 23654 brres2 35689 inxpss 35729 inxpss3 35731 idinxpssinxp2 35735 inxp2 35779 inxpxrn 35803 |
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