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Mirrors > Home > ILE Home > Th. List > relxp | GIF version |
Description: A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
relxp | ⊢ Rel (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 4707 | . 2 ⊢ (𝐴 × 𝐵) ⊆ (V × V) | |
2 | df-rel 4606 | . 2 ⊢ (Rel (𝐴 × 𝐵) ↔ (𝐴 × 𝐵) ⊆ (V × V)) | |
3 | 1, 2 | mpbir 145 | 1 ⊢ Rel (𝐴 × 𝐵) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2722 ⊆ wss 3112 × cxp 4597 Rel wrel 4604 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-in 3118 df-ss 3125 df-opab 4039 df-xp 4605 df-rel 4606 |
This theorem is referenced by: xpiindim 4736 eliunxp 4738 opeliunxp2 4739 relres 4907 codir 4987 qfto 4988 cnvcnv 5051 dfco2 5098 unixpm 5134 ressn 5139 fliftcnv 5758 fliftfun 5759 opeliunxp2f 6198 reltpos 6210 tpostpos 6224 tposfo 6231 tposf 6232 swoer 6521 xpider 6564 erinxp 6567 xpcomf1o 6783 ltrel 7952 lerel 7954 fisumcom2 11369 fprodcom2fi 11557 txuni2 12823 txdis1cn 12845 xmeter 13003 reldvg 13215 |
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