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Mirrors > Home > ILE Home > Th. List > brxp | GIF version |
Description: Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
Ref | Expression |
---|---|
brxp | ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3930 | . 2 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
2 | opelxp 4569 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
3 | 1, 2 | bitri 183 | 1 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 〈cop 3530 class class class wbr 3929 × cxp 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 |
This theorem is referenced by: brrelex12 4577 brel 4591 brinxp2 4606 eqbrrdva 4709 xpidtr 4929 xpcom 5085 tpostpos 6161 swoer 6457 erinxp 6503 ecopover 6527 ecopoverg 6530 ltxrlt 7830 ltxr 9562 |
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