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Mirrors > Home > ILE Home > Th. List > relco | GIF version |
Description: A composition is a relation. Exercise 24 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) |
Ref | Expression |
---|---|
relco | ⊢ Rel (𝐴 ∘ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4401 | . 2 ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} | |
2 | 1 | relopabi 4512 | 1 ⊢ Rel (𝐴 ∘ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ∃wex 1422 class class class wbr 3806 ∘ ccom 4396 Rel wrel 4397 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2612 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-opab 3861 df-xp 4398 df-rel 4399 df-co 4401 |
This theorem is referenced by: dfco2 4871 resco 4876 coiun 4881 cocnvcnv2 4883 cores2 4884 co02 4885 co01 4886 coi1 4887 coass 4890 cossxp 4894 funco 4991 fmptco 5383 cofunexg 5790 dftpos4 5933 |
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